Natural Advantage, Location and Trade Patterns in Increasing Returns to Scale Industries

Minerva, Gaetano Alfredo Working Paper Natural Advantage, Location and Trade Patterns in Increasing Returns to Scale Industries Quaderni - Working Paper DSE, No. 560 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Minerva, Gaetano Alfredo (2006) : Natural Advantage, Location and Trade Patterns in Increasing Returns to Scale Industries, Quaderni - Working Paper DSE, No. 560, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4727 This Version is available at: https://hdl.handle.net/10419/159401 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/3.0/ Natural Advantage, Location and Trade Patterns in Increasing Returns to Scale Industries ∗ Gaetano Alfredo Minerva † Dipartimento di Scienze Economiche, Universit`a di Bologna, Strada Maggiore 45, 40125 Bologna, Italy This version: February 2006 Abstract In a two sectors, two regions economy I show that the higher increasing returns to scale of an industry, the easier it will concentrate in response to natural advantage. To this end, one sector is assumed to be perfectly competitive and the other is monopolistically competitive, with a region’s firms producing at a lower marginal cost than the others in the monopolistic sector (or equivalently producing varieties more intensely demanded by consumers). If capital is mobile between regions in the long run, I analytically characterize the process of industrial location of the imperfectly competitive sector in the region with the comparative advantage. Keywords: Industrial location; monopolistic competition; intraindustry trade; cost advantage; demand intensity. JEL Classification: D43; F12; F21; L13; R12. ∗This paper is a revised version of Discussion Paper n. 40/2004 of the Department of Economics of the University of Pisa. I would like to thank for many helpful suggestions Gianmarco Ottaviano, participants to a lunch seminar in Bologna, to the 1st doctoral students Seminar held by Societ`a Italiana di Economia e Politica Industriale at the University of Ferrara, and to the 6th European Trade Study Group Conference at the University of Nottingham. I am greatly indebted to Kristian Behrens for his careful reading of an earlier draft and very useful remarks. †Telephone: +39 051 2092641. Fax: +39 051 2092664. E-mail: [email protected] 1 1 Introduction This paper follows a well established tradition in economic theory by studying the interaction between increasing returns and local market conditions in the joint determination of trade patterns and firms’ location. The main issue this paper addresses is to establish how exogenous variations in increasing returns to scale map into changes of location equilibria and, consequently, trade patterns in a two regions world, where one of them has a comparative advantage over the other in the production of imperfectly competitive commodities. Since the comparative advantage descends from an exogenous technological difference in the increasing returns to scale industry, I will refer interchangeably to the terms natural advantage and comparative advantage. I will argue, in general terms speaking, that the higher market power of firms in a given industry, the more sensitive they will be to local market conditions. Local markets’ attractiveness can be defined with respect to several dimensions: for instance, it could be defined in terms of the size of local demand, as in Krugman (1980) and Venables (1987) trade models, or in terms of a local cost advantage, within a one-factor-of-production Ricardian model, as in Venables (1987) again. In investigating how location and trade depend on the intensity of scale economies, when a region has a natural advantage that leads to a comparative advantage, I then extend Venables research. Let us set up the model. Economic space is made of two regions hosting two sectors which differ in the underlying market structure. The first is monopolistically competitive and produces an array of horizontally differentiated varieties, while the other is a residual sector, characterized by perfect competition, representing the rest of the economy. I assess the emergence of different patterns of trade and location equilibria in the monopolistic sector as a consequence of: a) a differential in marginal cost among firms according to the region they belong to; b) a differential in the degree of competition in each region, due to the difference in the number of firms in each local market; c) different degrees of overall competitive pressure, measured by the total number of firms in the economy, determined in turn by the degree of increasing returns to scale; d) the abandoning of 2 the CES (Dixit-Stiglitz) monopolistic competition model, in favour of a linear demand specification. Due to the linearity of the demand for differentiated varieties, we will show that point a) is amenable to an interpretation in terms of different intensities of demands for the differentiated products, according to the region where they are manufactured. The perfectly competitive sector is characterized by constant returns to scale. The monopolistically competitive sector is modelled according to a quadratic specification cast in an economic geography setting by Ottaviano, Tabuchi, and Thisse (2002). This sector can be thought to be manufacturing. The absence of strategic interaction among firms makes monopolistic competition particularly suited to capture market structure prevailing in traditional sectors as textiles, clothing, and food processing. After having derived short-run trade equilibria, we make the hypothesis that capital is mobile between regions in the long run, and flows where the rental rate is higher. I build on Ottaviano et al. (2002) and Behrens (2004, 2005) modelling. While their papers are full-fledged core-periphery (CP henceforth) models (what is mobile there are workers that locate where indirect utility is higher) I assume the mobility of capital towards locations where the rental rate is higher. The main contribution of the present paper is to add to the picture exogenous asymmetries between the two regions, and to see how they interact with increasing returns to scale in shaping the space economy. I model asymmetries as stemming either from a cost advantage of producing in region Aover region B, or from a demand premium that varieties manufactured in Aenjoys with respect to Bproducts. The linearity of the demand functions then ensures that, from the point of view of the individual firm, profit functions in the two circumstances are analytically equivalent, giving to our problem a twofold interpretation. The basic set up employed in this paper then follows what is known in the economic geography literature under the headings of footloose capital model (Martin and Rogers, 1995, Ottaviano, 2001, Baldwin et al., 2003), FC model hereafter. The model choice is dictated by the need of building a framework not too far from standard trade models, such that labour is immobile but capital and firms are mobile, thus easing the comparison of 3 our results to those of the trade literature. The differences between CP and FC models are thoroughly analyzed in Baldwin et al. (2003). For our purposes it suffices to stress the following. In a CP setting the mobile factor is labour, so that its migration induces also an expenditure shifting in the region where migration occurs, because people spend their earnings where they live. On the contrary, in a FC model the mobile factor is capital, whose rewards are repatriated to capital owners who are immobile in the two regions. Due to this fundamental difference, the FC model does not show circular causality (selfreinforcement) in agglomeration, and is more tractable analytically.1 As already shown in Behrens (2005), the linear demand model gives rise to asymmetric trade patterns when the two regions differ in the number of firms located, with one region hosting significantly more firms than the other. The idea is that when many firms are located in a region it is difficult to penetrate its market. The possibility of asymmetric trade patterns is a realistic feature being inevitably lost under the CES Dixit-Stiglitz specification when both regions host a positive share of firms. Introducing cost (or demand) asymmetries among the two regions, so that the more crowded region produces at a lower cost, strengthens the tendency of asymmetric patterns to arise. If they want to export to the other region, firms in the high cost one have to overcome the cost disadvantage as well. This makes the range of two-way trade smaller. As argued in Behrens (2004), the choice of including or not in the computation of the price index of a given region varieties not traded in equilibrium is sometimes essential for the resulting equilibrium. This is precisely the case under the weighted quadratic utility specification of Tabuchi and Thisse (2002). If different ways of computing the price index in a given region yield different results, the equivalency between price and quantity competition should a fortiori be invalid in trade models with non-traded varieties, and this is a striking difference with respect to baseline specifications of monopolistic competition models. I argue that, in Behrens model, price competition and a special 1Another possible comparison could be carried out between the FC model and the footloose entrepreneur model (see Baldwin et al., chapter 4). I prefer to stick to the comparison with the CP model since previous literature on asymmetric trade in economic geography was about CP models. 4 assumption he makes about prices of non-traded commodities are indeed equivalent to assume quantity competition. This feature gives to Behrens conjecture more strength with respect to competing hypotheses about the formation of the price index. All subsequent calculations in this paper are derived consistently with quantity setting, since it simplifies computations. The paper is organized as follows. In section 2 we present the model. In section 3 we compute the short-run equilibrium of the economy (for a fixed spatial distribution of firms) distinguishing among different trade patterns. Afterwards (section 4) we let capital going where the rental rate is higher. In some cases, which we analytically characterize, full agglomeration of the manufacturing sector in one region will be the long-run equilibrium. Section 5 focuses on the link between location, trade surplus, and scale economies in the presence of natural advantage. 2 The model The model developed in this paper is in various manners linked to other works in the economic geography literature. The closest relatives are Belleflamme et al. (2000), Ottaviano et al. (2002), and Behrens (2004, 2005). The economy is made of two regions s={A, B} of equal size, and two sectors: a monopolistically competitive sector, producing an array of differentiated varieties, and a perfectly competitive sector producing a homogeneous good 0, which we may think of as a composite commodity summarizing the rest of the economy. 2.1 Consumer’s behaviour The representative consumer in the two regions shares the same preferences and maximizes the following utility function: U(q0, x(j)) = ξZj∈N x(j)dj −1−ω 2Zj∈N [x(j)]2dj −ω 2Zj∈N x(j)dj2 +q0(1) 5 Parameters in the utility function are ξ > 0 and 0 ≤ω≤1. The set of varieties is S={j|j∈[0, N]}. They are uniformly distributed on [0, N], with Nbeing the total mass of the monopolistic sector.2The parameter ξis a proxy for the intensity of preference for the differentiated good. The higher ξ, the higher this preference. The parameter ω represents the degree of product differentiation among varieties. When ωapproaches zero varieties are so much differentiated that they can be thought to belong to completely different sectors (total utility is simply additive in the utility derived from each good and the inverse demand function of each variety, derived later, only depends on the quantity demanded of that same variety), while ωequal to 1 represents perfectly homogeneous products. We assume that the representative consumer in region sis endowed with Ksunits of capital and Lunits of labour, with labour supply Lbeing equal in the two regions. Income comes from the rental rate of capital and wage. The budget constraint of the representative individual in region Acan be written as Zj∈nA p(j)x(j)dj +Zj∈nB p(j)x(j)dj +p0q0=wAL+ max{rA, rB}KA(2) where p(j) is the price of a variety, x(j) is the quantity demanded, wAis wage in region A,rAis the rental rate of capital in region A, and rBrental rate in region B. Consumers will allocate capital in the region where the prevailing return is higher. We distinguish between varieties produced in region A(whose mass is nA), and varieties produced in region B(whose mass in nB). The quasilinear structure of U(·) implies that consumption of commodity 0 is the residual of what is spent on the monopolistic sector. Consequently, provided income is high enough so to allow a positive consumption of good 0 in equilibrium, every further increase in income corresponds to an equal increase in the consumption of the agricultural commodity, not affecting the demand for the differentiated varieties in manufacturing. 2In Vives (1990) and Belleflamme et al. (2000) the total mass of the monopolistically competitive sector Nis normalized to 1. We do not use this normalization because Nwill turn to be one of the key parameters of the model. 6 After having plugged the budget constraint in the utility function, maximization yields inverse demand functions. Inverse demand for a variety j∈nAproduced in Aand sold in Ais pAA(j) p0 =ξ−(1 −ω)xAA(j)−ωXA(3) where xAA(j) is demand for variety jand XA=Zj∈nA xAA(j)dj +Zj∈nB xAB(j)dj is total demand for the monopolistic sector from consumers located in region A, consisting of varieties manufactured both in region Aand in region B. A variety j∈nBproduced in Bbut sold in Ahas an inverse demand equal to pAB(j) p0 =ξ−(1 −ω)xAB(j)−ωXA(4) where variables have the same interpretation as above. Similar expressions can be derived for products sold in market B. 2.2 Labour market Lis employed as a variable input either in manufacturing or in agriculture, and the supply of labour is perfectly elastic between sectors. In manufacturing, csunits of labour are needed for each unit of output, and this labour requirement differs in the two regions, that is cA6=cB. This assumption wants to capture the fact that there are locations where productivity of labour in manufacturing is higher. Turning to the production of the homogeneous good, it is carried out under constant returns to scale, with unit labour requirement equal across the two regions and set equal to one by an appropriate choice of scale. A positive amount of labour is employed in sector 0 because labour supply is assumed to be high enough so to cover all input requirements of the differentiated commodity sector. Constant returns to scale ensure that wage in sector 0 is equal to the exogenously fixed price p0. Since labour market is assumed to be in equilibrium, wA=p0. If wA≷p0workers would move from one sector to the other until equality in the wage 7 rates is reached due to perfect elasticity of supply. Total numeraire production Q0in region Ais then Q0≡L−Zj∈nA cAx(j)dj. 2.3 Firms’ behaviour Firms play a two-stages game. In the first stage they establish in each region, up to the point clearing capital market. Each plant requires φunits of capital for functioning. By an appropriate choice of scale, each plant’s capital requirement can indeed be normalized to one (entry cost). In this case, the mass of firms will exactly equal the mass of capital available in that region at a given moment.3In the second stage there is market competition. As said earlier, csunits of labour are needed to produce one unit of the differentiated output, and this marginal cost differs across regions. Markets are segmented, so that each firm sets the strategic variable (price or quantity) in each regional market in which it operates. Notice that exporting in the foreign region requires tunits of good 0 for each unit of output. Total profits of a representative Afirm are then: ΘA(i) p0 =pAA(i) p0 −cAxAA(i) + pBA(i) p0 −cA−txBA(i)−rA p0 We substitute inverse demands (3) and (4) in the profit function. We do so because we assume that firms maximize profits with respect to quantities. As claimed by Vives (1990), in a model of monopolistic competition maximization with respect to prices or quantities brings the same results, since the individual firm behaves as a monopolist on the residual demand. The equivalency could possibly fail in a trade model like ours if trade does not take place actually, because transport costs are too high, and a positive foreign demand does not correspond to a price greater or equal to costs. In this case firms may be thought to set a fictional price abroad, even if demand at this price is zero, and this 3This is the same assumption made in Martin and Rogers (1995). As they do, we will introduce two different time horizons. In the short run capital available in each region equals the capital endowment of the representative consumer in that region (Ks=ns). In the long run capital flows freely from one region to the other, so that the only equality that has to hold is KA+KB=N. 8 exceeds 2NBB, the fact that xBB be positive is not compatible with the assumption that λ > 1/2. Firms in Bcannot export a positive quantity to Aas long as t>η−θ. If t<η−θ, x∗ AB >0 provided λ < 2(η−θ−t)(1 −ω) ωN(θ+t)≡νAB (10) We have that νAB ∈(1/2,1) when NAB < N < 2NAB where NAB ≡2(η−θ−t)(1 −ω) ω(θ+t) with νAB >1 for N < NAB (νAB >1/2 for N < 2NAB). Two cases should be distinguished at this point: the cost advantage of region Acould be high (respectively low), if θ > η−θ(respectively θ < η −θ). In terms of asymmetries of the demand functions, the intercept of the demand for Bproducts η−θcould be smaller (respectively bigger) than the difference θbetween the two intercepts. In what follows we stick to the following assumption. Assumption 2. The cost advantage θis such that θ < η −θ. The analysis could be carried out without substantial changes to the results for the case θ > η −θas well, but for simplicity it is carried out only in one case. Moreover, if we interpret the model as one featuring taste asymmetries, it is preferable to assume that the difference between the intercepts of the demand functions (θ) be smaller than the smallest intercept (η−θ). First we consider autarchy, the case involving no-trade among the two regions. Lemma 1. Autarchy constitutes the short-run equilibrium if one of the following conditions is satisfied: i) t > η; ii) θ < t < η, with N > NBA and λ≤νBA. 15 Proof. Point i) is easily derived. As to point ii), autarchy is the short-run equilibrium only if N > NBA, that is only when x∗ BA could be zero. Both for θ < η −θ < t < η, and for θ < t < η −θ < η, this is true if λ≤νBA and N > NAB. Since 2NAB < NBA, when N > NBA x∗ AB is zero. Under one-way trade firms in Asupply domestic and foreign markets, while firms in Bsupply their domestic market only. There is an asymmetry in trade relations. Lemma 2. One-way trade constitutes the short-run equilibrium if one of the following conditions is satisfied: i) for η−θ < t < η,N < NBA; or N > NBA and λ>νBA; ii) for θ < t < η −θ,NAB < N < 2NAB and λ≥νAB; or 2NAB < N < NBA; or N > NBA and λ>νBA; iii) for t < θ,NAB < N < 2NAB and νAB ≤λ < νBB; or 2NAB < N < 2NBB and λ<νBB. Proof. Let us start from t > θ. Remember again that NBA >2NAB. Then simply consider all the combinations of Nand λensuring that x∗ BA >0 and x∗ AB = 0. When t < θ, it is possible to show that NBB > NAB. Nothing can be said about the ordering among NBB and 2NAB and the threshold νBB becomes redundant when it is greater than 1 (think for example of a case where 2NAB < N < NBB). We now characterize two-way trade, when both regions trade with each other. Lemma 3. Two-way trade constitutes the short-run equilibrium for t < η −θ, and N < NAB; or NAB < N < 2NAB and λ<νAB. Proof. We derive conditions making the quantity sold abroad by Bfirms, x∗ AB, positive either for θ < t < η −θ, or t < θ < η −θ. Two-way trade is possible for every admissible λwhen N < NAB. When the total mass is NAB < N < 2NAB, for two-way trade to be possible it has to be λ<νAB. As to the quantity x∗ BB, it will be always positive under the conditions stated in the proposition: it suffices to remind that NAB < NBB (implying trivially 2NAB <2NBB) and νAB < νBB. 16 As mentioned earlier, the role played by N, and scale economies has often been neglected in the literature. Notice that when NAB < N < 2NAB, the share of firms located in Ashould not exceed the threshold νAB if we want two-way trade to be feasible. If this were not the case, then it would be prohibitive to export to region Afor Bfirms, due to toughness of competition. If the total mass of the monopolistic sector exceeds 2NAB, two-way trade is impossible, since profits’ margins will be compressed by the large number of firms, and Bfirms will not be able to export under the assumption that λ > 1/2.9 Belleflamme et al. (2000) in their paper restrict attention to two-way trade. The only condition imposed concerns the level of transport costs t, that should be sufficiently low, and they normalize the total mass Nto 1. By Lemma 3, the relative share λcould be ignored only if N < NAB (which corresponds to NAB >1 in their setting). If NAB < N < 2NAB, as the agglomeration process of firms in region Aunfolds, when λreaches νAB two-way trade is no longer sustainable, and the short-run equilibrium consists of one-way trade.10 We say that in region Ba process of deindustrialization has occurred when it is not possible for a firm operating in Bto make non-negative profits. Lemma 4. Short-run equilibrium involves deindustrialization of region Bfor t<θ, NBB < N < 2NBB and λ≥νBB; or N≥2NBB. To appreciate the economic meaning, let us focus on the cost of products that could be sold in market B. The fact that t < θ, means that the total cost for Afirms (cA+t) of a product sold in region Bis lower than the cost incurred by Bfirms themselves (cB). If N≥2NBB, only Afirms are capable of being in the market, assuming a spatial distribution λ∈(1/2,1). 9When N≥2NBB , firms in Bcannot profitably produce even for their domestic market, see below Lemma 4. 10Even if our model is not equivalent to Belleflamme et al., because we assume that the cost differential is fixed, what they do is hence to assume implicitly that N≡1< NAB , NAB >1⇔2(η−θ−t)(1 −ω)> ω(θ+t) which is an additional parameters’ restriction that should be indicated explicitly. 17 Equilibrium prices can be derived simply substituting equilibrium quantities in (3) and (4). Conditions for the non-negativity of mark-ups (p∗ AA −cA) and (p∗ BA −cA−t) coincide with those for equilibrium quantities x∗ AA,x∗ BA so that non-negativity of mark-ups is implied by non-negativity of quantities (the same applies for firms in B). 4 Long-run equilibrium Bidding for available capital determines the equality between equilibrium operating profits, Π∗ s, and the rental rate in the short run, r∗ s/p0= Π∗ s, for a given spatial distribution of firms.11 In the long run capital is mobile between regions so that the spatial distribution of firms is no longer equal to the initial endowment of capital in Aand B. Capital flows occur in response to the differential in the equilibrium rental rate r∗ A(λ)−r∗ B(λ), determined in the short-run. When the differential is positive capital goes from region Bto region A. Viceversa, when the differential is negative, capital flows out of Ainto B. For every trade pattern we identified, we argue about existence, uniqueness and convergence to the equilibrium distribution λ∗, so that rA(λ∗) = rB(λ∗), in the sections below. If an interior equilibrium is reached, operating profits in the two regions will be equalized as well, Π∗ A= Π∗ B. In other terms, individuals look first for investment opportunities in the local market (which fixes the rental rate at operating profits in that region), then they look abroad, causing the exit of firms in the local market and the subsequent entry in the foreign one if the rental rate obtained there is higher (which determines equality of rental rates and operating profits across regions). In what follows we compare operating profits in the two regions under different trade patterns. Our goal is to establish which firms are performing better, whether those located in Aor in B, as a function of the total mass of the monopolistic sector N, and the spatial distribution λ. Once having obtained these results, it is possible to determine the long-run outcome of the economy, under the assumption that capital is invested where interest rate 11Competition in capital market drives the rental rate down to operating profits because of free entry of firms hiring capital. 18 is higher. 4.1 Autarchy Under autarchy, operating profits of a variety i∈[0, ns] produced in s∈ {A, B}are Πs(i) = [ξ−(1 −ω)xs(i)−ωXs−cs]xs(i) where total output under autarchy in region sis equal to Xs=Zj∈ns x(j)dj Equilibrium output x∗ sis: x∗ s=ξ−cs 2(1 −ω) + ωns (11) and correspondingly equilibrium price is p∗ s p0 =(1 −ω)(ξ+cs) + csnsω 2(1 −ω) + ωns Profits are finally Π∗ s=(1 −ω)(ξ−cs)2 [2(1 −ω) + ωns]2(12) Proposition 1. When differentiated varieties are not traded (see Lemma 1), Π∗ A>Π∗ B if: i) N < NU, where NU≡2θ(1 −ω) ω(η−θ) and every λ; ii) N > NUand λ<λU, where λU≡2θ(1 −ω) + ωηN ωN(2η−θ) with λUbeing the long-run spatial equilibrium. Proof. Solving the inequality Π∗ A>Π∗ Bleads to the condition ω[(η−θ)nA−ηnB]<2θ(1 −ω) 19 that, after having substituted nA≡λN, and nB≡(1 −λ)N, is equivalent to λ < 2θ(1 −ω) + ωηN ωN(2η−θ)≡λU(13) When λ=λUit immediately descends that Π∗ A= Π∗ B. It is easy to argue about stability of the long-run equilibrium, since (12) is monotonically decreasing in ns. Assuming that N < NU, where NU=2θ(1 −ω) ω(η−θ) makes λUbigger than 1, so that Π∗ A>Π∗ Bfor every λand the equilibrium involves full agglomeration of the manufacturing sector in region A, provided conditions in Lemma 1 are fulfilled. The proposition says that a long-run equilibrium involving partial agglomeration exists only if the total mass of the monopolistic sector is greater than NU. The economic intuition of this result is the following. When the total mass of firms Nis small, and scale economies are strong, firms in Awill make higher profits for every admissible λ, because of the cost advantage θ: region Ais sufficiently attractive to host the whole manufacturing sector. On the other hand, if Nis large, then the whole manufacturing sector could not locate entirely in A, still doing better than an isolated firm in B. In this case the actual spatial distribution of firms will matter for profitability, and the fraction of firms in Ashould be small enough to get Π∗ A>Π∗ B. Two components related to the degree of competition affect profitability: the first is overall competitive pressure, measured by N, being a measure of increasing returns to scale; the second is local competitive pressure, measured by λ. The bigger the cost advantage θ, the larger the values of N, and the lower scale economies intensity, under which we get full agglomeration. Similarly to NU, also λUincreases as θrises. A partially agglomerated stable spatial equilibria under autarchy is also found in Behrens (2004). We now proceed to show how the same kind of reasoning applies to the other trade regimes. 20 4.2 One-way trade Under one-way trade, firms located in region Bdo not make positive export to A. The output index in region Ais then XA=Rj∈nAxAA(j)dj. Substituting in (6), and employing the symmetry of the model, the equilibrium quantity x∗ AA is x∗ AA =η 2(1 −ω) + ωnA equal to (11), the quantity sold in region Aunder autarchy. As in a situation without trade at all, firms in Aare protected against competition coming from foreign firms, and they behave in the same way of autarchy in the local market. This makes the home component of profits equal to autarchy profits. Profits of Afirms are also made of a component coming from abroad, making total profits equal to: Π∗ A= Πh A+ Πf A=(1 −ω)η2 [2(1 −ω) + ωnA]2+[2(η−t)(1 −ω) + ωnB(θ−t)]2 4(1 −ω)(2 −2ω+ωN)2.(14) This is the sum of home profits (Πh A) and foreign profits (Πf A). Profits of firms in Bare Π∗ B= Πh B=[2(η−θ)(1 −ω)−ωnA(θ−t)]2 4(1 −ω)(2 −2ω+ωN)2(15) corresponding just to the home component. A sufficient condition for Aprofits to be greater than Bprofits is t < θ, which turns to be true under case iii) of Lemma 2. If the cost (or demand intensity) advantage of region Ais greater than transport costs, markets are relatively well integrated and location in A allows higher profits regardless of the spatial distribution. When t>θ, I am not able to provide a closed-form solution for λO, the long-run spatial equilibrium such that Π∗ A(λO)=Π∗ B(λO), and Π∗ A(λ)≷Π∗ B(λ) for every λ≶λO. Nonetheless in Appendix 7.2 I prove that, whenever this value exists, it is unique. Results are summarized in the proposition that follows. Proposition 2. When one-way trade is established (see Lemma 2), Π∗ A>Π∗ Bif t < θ. If t > θ, we have one of the following cases: i) If N < NO, where NO≡2(1 −ω) ω(t−θ)pη2+ (η−t)2−η+θ 21 Π∗ A>Π∗ Bfor every λ. ii) If N > NO, and the long-run spatial equilibrium λOexists, then there exists a unique λO, and Π∗ A≷Π∗ Bfor λ≶λO. iii) If N > NO, and λOdoes not exist, then Π∗ A<Π∗ B. Proof. See Appendix 7.2. Under one-way trade and t>θ, several configurations are possible in the short run, with firms in Aperforming better than firms in B, or, viceversa, firms in Bdoing better than in A. If λOdoes not exist, and in the Appendix I state when this is the case, one-way trade cannot be a stable long-run outcome of the economy: we have either full agglomeration (point i) in the proposition) or transition to another trade pattern (point iii)). If λOexists, then the interior equilibrium will be stable. Behrens (2005) shows that partial agglomeration is unstable as soon as one-way trade emerges. This contrasts our result about the stability of λO, provided its existence. The reason lies in the fact that in this paper the only mobile factor is capital, while consumers-workers stay put in their regions of residence. In other words, our model lacks the strong agglomeration forces typical of the core-periphery setting.12 In the CP model of Behrens (2005) migration of workers generates both demand-linked circular causality (migration generates expenditure shifting by workers, which generates in turn production shifting, and this determines more migration to fulfill firms’ fixed costs requirements) and cost-linked circular causality (a higher mass of differentiated products is available where production is concentrated, and workers find more convenient to locate there to save on trade costs), whereas in our FC model, these effects are not present. The existence of a stable λOin our framework highlights once more that the type of spatial equilibrium depends on the type of factor that moves across regions, and it makes a difference assuming that capital, instead of labour, is mobile. 12See Chapter 3 in Baldwin et al. (2003) for a comparison of FC and CP models. 22 4.3 Two-way trade From Lemma 3, two-way trade is possible only if t<η−θ. At the same time, the total number of firms in the economy has to satisfy the conditions N < NAB; or NAB < N < 2NAB, and λ<νAB. Profits of firms in Aare Π∗ A= Πh A+ Πf A=[2η(1 −ω) + ωnB(θ+t)]2+ [2(η−t)(1 −ω) + ωnB(θ−t)]2 4(1 −ω)(2 −2ω+ωN)2(16) while profits of firms in Bare Π∗ B= Πh B+ Πf B=[2(η−θ)(1 −ω)−ωnA(θ−t)]2+ [2(η−θ−t)(1 −ω)−ωnA(θ+t)]2 4(1 −ω)(2 −2ω+ωN)2 (17) made up of a home component and a foreign component. The following proposition explains the relative profitability of firms in the two regions as a function of the total mass Nand the share λ. Proposition 3. When two-way trade is established (see Lemma 3), Π∗ A>Π∗ Bif t < θ. If θ < t < η −θone of the following conditions has to be satisfied: i) N < NT, where NT=2θ(1 −ω)(2η−t−θ) ω(θ2+t2) and every λ; ii) N > NTand λ<λT, where λT≡1 2+1 2 NT N with λTbeing the long-run spatial equilibrium of the economy. Proof. If t<θ, it is easy to see that Π∗ A>Π∗ B. If θ < t < η −θ, comparing (16) and (17), Aprofits are greater than Bprofits if nA−nB<2θ(1 −ω)(2η−t−θ) ω(θ2+t2)≡NT(18) which could be expressed in terms of λand Nas λ < 1 2+1 2 NT N≡λT 23 The threshold λTis lower than one as long as N > NT. The stability of the long-run equilibrium in this case is ensured by the fact that profits are monotonic in λunder twoway trade. When N < NTwe have full agglomeration in A, provided conditions in Lemma 3 are fulfilled. If the total mass of firms is big enough, this guarantees the existence of a spatial distribution making better off firms in Bin the short run. The long-run behaviour of the economy depends as usual on the assumption that capital flows where the interest rate is higher, with the interest rate equal to operating profits, and a long-run equilibrium with partial agglomeration and bilateral trade exists. This is not the case in the CP setting of Ottaviano et al. (2002), where the equilibrium is either symmetric or involves full agglomeration in one region. As in the previous trade patterns, the partially agglomerated nature of the equilibrium does not depend on the cost asymmetry only. Keeping fixed θ, depending on N, so on the strength of returns to scale, we have different spatial configurations. The effect played by Nis similar to the one present in CP models: a lower N(higher increasing returns to scale) makes full agglomeration more likely. The explanation is partially different. In CP models stronger scale economies (higher fixed costs) imply that firms’ relocation will involve a considerable amount of demand shifting, due to workers’ migration, this enhancing circular causality. Since in a FC model there is no demand shifting, our context retains just one effect of returns to scale: the higher fixed costs, the lower the total mass of firms, the lower overall competitive pressure, the higher mark-ups and profits, so that firms more easily establish in the region with the location advantage because they are influenced less intensely by the presence of the other competitors. 4.4 Full versus partial agglomeration in the long run We now characterize in terms of the parameters’ values, and in terms of the total mass of the monopolistically competitive sector the emergence of full agglomeration of manufacturing in region A. We give conditions so that, starting from a short-run equilibrium 24 be achieved directly making the behavioural assumption that firms set quantities. Actually in this case the dependent variable in the demand functions has to be p(i). The perceived demand is p(i) = ξ−(1 −ω)x(i)−ωZN 0 x(i)di while the realized demand is p(i) = ξ−(1 −ω)x(i)−ωZN 0 x(i)di +µ(i) It is straightforward to see that the two demand functions always coincide as long as µ≡0, which is a necessary condition to get equivalency between the two optimization problems (if µ(i)6= 0 for some ithe equivalency never holds). Moreover when µ≡0 conditions (21) and (22) reduces to (19). 7.2 Proof of Proposition 2 Step 1 (Non-monotonicity of Π∗ A(λ)). We substitute in (14) and (15) the expressions nA≡λN, and nB≡(1 −λ)N. First of all we determine whether, under one-way trade and t>θ, Π∗ A(λ) and Π∗ B(λ) are strictly monotonic in λ. It is easy to see that Π∗ Bis increasing in λ. The function Π∗ A(λ) is non-monotone. First of all notice that ∂Πh A/∂λ < 0. Then we have that ∂Πf A/∂λ ≥0 (both quantity x∗ BA and price p∗ BA are non-decreasing in λ). Moreover ∂Πf A(λ, t)/∂λ = 0 when t={θ, tsup}, where tsup is the maximum value of transport costs compatible with one-way trade for a given λ(derived making explicit in (8) transport costs t). The function ∂Πf A(λ, t)/∂λ has a unique maximum in t, computed equalizing to zero its derivative, let it be tmax. Then if ∂Πh A(λ) ∂λ +∂Πf A(λ, t) ∂λ t=tmax <0,(23) ∂Π∗ A(λ, t)/∂λ < 0 for every admissible t. Actually it turns out that (23) is less than zero if and only if the following condition is verified: η2 [2(1 −ω) + ωλN]3>(η−θ)2 4(2 −2ω+ωN)2[2 −2ω+ω(1 −λ)N](24) 31 Consequently (24) does not hold when λis sufficiently close to one and θis small. In such a case profits of firms located in region Aincrease as the share of firms in Aincreases, because the rise in profits coming from the foreign region more than offset the fall in the home component. The function Π∗ Acan be first decreasing and then increasing in λas it tends to 1, provided that tis in a neighborhood of tmax. Step 2 (Uniqueness of λO). We demonstrate the following two properties. They turn to be useful when dealing with existence and uniqueness of λO. Property P1. The first is that ∂Π∗ B(λ, t) ∂λ λ=1 −∂Π∗ A(λ, t) ∂λ λ=1 >0.(25) Computing (25), we get the following condition: Nωφ(ω) 2(1 −ω)(2 −2ω+ωN)3>0, where φ(ω) is a parabola with upward concavity and imaginary roots, so that it is always positive. Property P2. The second property we are interested in is that ∂2Π∗ A/∂λ∂λ > 0, meaning that Π∗ Ais a convex function. Taken together, these two properties ensure that whenever Π∗ B(λ) crosses Π∗ A(λ) it will do it only once: provided λOexists in an admissible range of λ, it will be unique. Step 3 (Cases of non-existence of λO). Π∗ A(λ) and Π∗ B(λ) are continuous functions on λ∈(1/2,1], but existence of λOis not always guaranteed. The first case of non-existence is when Π∗ A(1) >Π∗ B(1). Given properties in Step 2, this is also a necessary and sufficient condition for Π∗ A(λ) to be greater than Π∗ B(λ) for every admissible λ. Solving the inequality, Π∗ A(1) >Π∗ B(1) if N < NO, where NO≡2(1 −ω) ω(t−θ)pη2+ (η−t)2−η+θ. Other cases of non-existence are when Π∗ B(λ) lies above Π∗ A(λ) for every admissible λ. In particular, it could be the case that, even though Π∗ A(λ) and Π∗ B(λ) intersect at some 32 λ∈(1/2,1], this point does not satisfy constraints νAB, or νBA under points i) and ii) in Lemma 2. When this is the case, in the long run we have transition from one-way trade to two-way trade (λ<νAB) or autarchy (λ≤νBA). 7.3 Proof of Proposition 5 I prove separately each point in the statement of the proposition. Point i). The proof descends from Lemma 1 and Proposition 1 and corresponds to full agglomeration with non-tradeable varieties. Point ii). If θ < t < η and we are in the short-run autarchic equilibrium (point ii) in Lemma 1), full agglomeration is not possible because no-trade requires that λ≤νBA <1, while full agglomeration obviously entails λ= 1. Full agglomeration cannot be reached unless transiting across the one-way trade short-run equilibrium. With one-way trade and η−θ < t < η (point i) of Lemma 2), Proposition 2 requires that N < NO. Notice that NO< NBA. This can be checked solving the corresponding inequality, and arriving at a point where it is straightforward to see that pη2+ (η−t)2<2η−t < 3η−t−θ When the total mass of firms is less than NOthen full agglomeration takes place. Point iii). When θ < t < η−θ, we could be either in a one-way or a two-way short-run equilibrium. Two-way short-run equilibrium occurs under conditions in Lemma 3. By Proposition 3, if N < NTthen Π∗ A>Π∗ Bfor every λ. If N < NAB, two-way trade is the short-run equilibrium for every share λ. We prove that whenever N < NAB we get full agglomeration, since NT−NAB >0. Solving this inequality is equivalent to solve f(θ)>0, where f(θ) is equal to f(θ) = (η−t)θ2+ 2tηθ −t2(η−t) (26) The function f(θ) is a parabola in θwith upward concavity, with two negative roots. Since all admissible values of θare greater than zero, f(θ) will be positive in this range, implying that NT−NAB >0. 33 For NAB < N < 2NAB, two-way trade arises only if λ<νAB, and full agglomeration in the long run can be reached only through transition to short-run one-way trade equilibrium (λ≥νAB, see point ii) in Lemma 2). With one-way trade, we recall that a necessary condition for complete agglomeration is N < NO. Transition to one-way trade happens if two-way trade is not a long-run equilibrium, which turns to be true in the following cases. The first case is when NAB < N < NT< 2NAB (equivalently NAB < N < 2NAB < NT), this making profits in Agreater than in Bfor every λunder two-way trade. Consequently, λrises until the economy experiences one-way trade. The second case is when NAB < NT< N < 2NAB, so that an equilibrium distribution λTexists. In such a case two-way trade equilibrium is impossible only if λT≥νAB. Again, there will be a switching to one-way trade before the equilibrium share λTcould be reached. Point iv). When t<θ, the short-run equilibrium depends on the total mass of firms N. Let us first consider short-run one-way trade (iii) in Lemma 2). Profits in Aare higher than in Bby Proposition 2. As capital moves to region A(λrises) we have deindustrialization of Bprovided N > NBB. Actually when λbecomes greater or equal to νBB profits of firms in Bare non-positive. Consequently all the residual capital in B is suddenly diverted towards region A, and this ensures complete agglomeration in A. If N < NBB, firms in Bmake positive profits for every λbut profits made in Aare higher and full agglomeration is attained again. 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J., 1987, Trade and Trade Policy with Differentiated Products: A Chamberlinian-Ricardian Model. Economic Journal,97, 700-717. [24] Vives X., 1990, Trade association disclosure rules, incentives to share information, and welfare. RAND Journal of Economics,21, 409-430. 37 - 6 Q Q Q Q Q Q Q Q Q Q Q Q Q Q p(i) ˜x∗(i)0 ¯p(i) Figure 1: The perceived demand function ˜x∗(i). 38