Endogenous mobility, human capital and trade István Kónya February 2, 2008 Abstract This paper presents a model that highlights the role o skilled migration in the emergence o regional dierences as a consequence o oreign trade. The model is based on the widely used increasing returns/transportation costs ramework, with heterogeneous households and imperect labor mobility added. When some regions have a geographical advantage in their access to the outside world, international trade leads to human capital reallocation. The paper s main contribution is to show that even small migration lows can lead to large inequalities in per capita incomes, i the most skilled workers move. JEL: F2, R2, R23 Keywords: Economic geography, migration, human capital, regional inequalities Introduction This paper is concerned with the role o trade opening and skilled migration in generating regional inequality. A model is presented in which skilled migration acts as a powerul magniication orce that ampliies regional dierences emerging rom regional dierences in market access. In particular, I study a scenario when an initially closed country opens up to international trade. I assume that the country s regions dier in their access to outside markets, which gives better located regions a natural advantage. In a closed economy, this advantage cannot be realized and spatial symmetry is an equilibrium. With external opening, however, the region closer to the outside market will have a higher real wage. This attracts immigrants rom the other parts o the country. In the presence o economies o scale, such migration leads to urther divergence. The crucial contribution o this paper is to endogenize not just the size, but also the composition o the migration low. Assuming that people dier more in their skills than in their migration costs, it is the most able who choose to move. This has two consequences or regional inequalities. First, in the presence o scale eects skilled migration leads to an increase in the wage rate o the immigrant region, and this increase will be larger than with homogeneous labor. The market size o the immigrant region will increase more than proportionately with the number o immigrants, because their income and supply o eiciency units o labor will be above average. Second, average incomes will be higher in the host region because o the composition eect. The immigrant region will have a distribution o skills skewed towards the highly skilled, whereas the opposite is true or the other region. Thus the model s prediction is that migration o skilled people reinorces natural advantages by both the scale and composition eects. There are many other models that generate regional inequalities. Papers in the "New Economic Geography" literature show how regional inequalities emerge as a result o endogenous See Fujita, Krugman & Venables (999) or an extensive overview o this literature.
orces (increasing returns and pecuniary externalities). In act, these models predict spatial asymmetry even when the underlying undamentals are the same. 2 In contrast, in this paper the primary cause o regional inequalities is exogenous. 3 Endogenous orces (increasing returns and migration), on the other hand, are mostly responsible or the extent o the inequalities. 4 This paper contributes to the literature by arguing that skilled migration is an essential ingredient to explain the magnitude o regional disparities. Earlier models either rely on largescale migration (Krugman 99) or a drastic change in the sectoral composition o regions (Krugman & Venables 995) to generate substantial regional inequalities. But in many cases, especially in Europe, we only observe small causes that lead to large income disparities. The model in this paper can oer an explanation or this phenomenon. Even i undamentals are not very dierent and the migration low is small, the resulting dierence in average incomes can be large, due to the scale and composition eects. As I show later, the composition eect is especially important quantitatively. 2 Model description Consider a world with two countries, Home and Foreign. Foreign is single geographic entity, but Home has two regions, the East and the West. Trade is always possible between the latter two, but Home is initially closed to trade with Foreign. The objective is to examine the impact o trade opening between the two countries. Figure : Geography o the three regions Trade is subject to transportation costs, which take the well-known "iceberg" orm. 5 For simplicity, the three regions are assumed to be located along a line (a linear world). As Figure below shows, the West is closer to Foreign than the East. The transportation cost parameter is τ FW between Foreign and the West, and τ WE between the West and the East. The linear world assumption guarantees that the transportation cost parameter between Foreign and the East is then τ FW τ WE. Initially Home s external transportation cost is given by τ FW =, while ater trade opening it is τ FW <. Regions are populated by households 6 with dierent amounts o human capital, which they rent out to irms who use it as a production input. Households can move between the East and the West, but rom Home to Foreign. Migration is subject to a ixed monetary cost D, which can be paid ater moving. This amounts to the existence o perect credit markets that can inance the cost o relocation. Regional human capital levels are denoted by H r. 2 An exception is Matsuyama (998), which has various core-periphery patterns emerging as a consequence o geographical dierences. 3 While the model is capable o generating endogenous agglomeration even in a closed economy, such an equilibrium is not inevitable - spatial symmetry remains an equilibrium as well. 4 There is some recent evidence that supports this view. Gallup, Sachs & Mellinger (998) document that areas with good access to transportation (coast lines, navigable rivers etc) are richer than other regions. In a study o Japanese regions, Davis & Weinstein (2002) conclude that locational undamentals determine basic concentration patterns, but endogenous agglomeration orces might play an important role in ampliying these patterns. 5 It is assumed that part o any good shipped simply "melts" in transportation. More precisely, i τ i j > units o a good are shipped rom region i, only unit arrives in region j. 6 Since households are composed o a single person, I will use the terms household, person, and worker interchangeably. 2
I normalize the total amount o human capital o Home to unity, i.e. H w + H e =. To simpliy notation, I will use H and H to indicate a particular human capital distribution across the two regions o Home. I assume that initial distribution o human capital (beore migration takes place) is symmetric between the East and West, with a common distribution h G(h); h 2 [0;] and aggregate regional levels o R 0 hdg(h) = =2. 2. Consumption Households in any region consume a variety o goods. There are a continuum o such goods, indexed rom 0 to N, where N will be determined endogenously. Consumers maximize the CES utility unction Z N u j = c j (i) = di ; () 0 where c j (i) is consumption o good i or household j, and is the constant elasticity o substitution between varieties. Goods enter the utility unction symmetrically, and consumers have a taste or variety 7. For the chosen market structure (see below) it is necessary to assume that >. Households earn income rom supplying human capital to irms in the region they live in. Units o human capital have a constant price w, thus household j with human capital h j receives an income o wh j. The budget constraint can be written as Z N 0 p(i)c j (i)di = wh j ; (2) with p(i) standing or the price o good i. 8 From () and (2) the demand unction o person j or good i is given as p(i) wh j c j (i) = P P : (3) Notice that since demand is linear in human capital, and prices and the rental rate or human capital are deined at the regional level, it is easy to ind aggregate demand or a region. Thus regional demand C r is identical to (3), except that h j is replaced by the total amount o human capital in region r, H r : pr (i) w r H r C r (i) = : (4) P r 2.2 Production P r The structure o production ollows the monopolistic competition model in Dixit & Stiglitz (977). Each variety is produced by a single irm that sets its own price, but an individual irm s decision does not aect the aggregate price index. Firms produce using human capital as input. Production requires α units o human capital independent o output, and β units o 7 This can be seen by setting c(i) = c=n, and noticing that the resulting expression is increasing in N, the measure o variety. 8 Although w and p(i) are region speciic, to save on notation, I omit regional subscripts whenever possible. 3
human capital per unit o output. Thus the cost unction is written as: TC = (α + βq)w; where q is the quantity produced by the irm. 9 Firms take the regional demands as given and set prices or each region they sell their products in. I assume that there are no arbitrage opportunities in trade, so that i a irm sets a price o p or its export good, the good will sell in the destination market or τ p, where τ is the "iceberg" transportation cost. Since the demand unction (4) has a constant price elasticity, optimal prices are a constant markup over marginal cost. Thus irms will set the same.o.b. (beore trade) price regardless o the destination market. Formally, substitute (4) or q into the proit equation, and rearrange the irst-order condition to get p = βw: (5) Since irms are symmetric, and marginal cost w is the same within a region, the same price is set or any variety within a region. To urther simpliy notation, we can choose units o goods so that β=( ) = and p = w. As the blueprints o an ininite range o varieties is assumed to be available, it is always possible to set up a irm producing a new variety. Free entry and the assumption that irms are ininitesimal drives proits to 0. Setting π = pq TC = 0 and rearranging, we get q = α: (6) Thus there is a unique scale o production where irms exactly break even. 2.3 Migration To anticipate results, I derive the condition or moving rom the East to the West. Person j moves i her utility is greater in the West. Given migration costs, her nominal income is h j D in the West and w e h j in the East. With homothetic preerences utility is proportional to the real wage, where the delator is the price index. Thus the condition or moving is given by h j D > w eh j w e P w ) h j > D: P w P e P e Notice that i it is proitable or person j to move, all other workers with human capital greater than h j will move as well. This is because the gain rom moving is linear in h j or individuals (see [??]), as their isolated actions do not inluence the aggregate wage rate and the migration cost is constant. However, the wage rate is bounded rom below (see next section), so gains rom migration are inite. I there are people with very low levels o human capital, there will always be someone or whom moving is not proitable. In what ollows, I will assume that this is indeed the case. 0 These two observations together imply that i there is migration in equilibrium, there must be a marginal person who is indierent between migrating or not. Let the human capital level o that person be h. Then every worker with h j > h will migrate, and all the others will stay. 9 Since the structure o irm decisions is the same across regions and products, I omit the regional and product subscripts when no conusion arises. 0 This assumption is not essential, but simpliies the analysis by ruling out ull agglomeration. 4
Thus the equilibrium condition or migration can be written as w e P w h D and h ; (7) P e with complementary slackness. 2.4 Equilibrium Factor markets clear in all regions. Using (6) and =( )β =, in region r we have H r = N r (α + βq) = N r α; which implies that the number o irms in region r is given as: N r = H r α : (8) To derive the market clearing conditions or goods, we can simpliy the price index P rom eq. (??) urther. To simpliy notation, let ρ = τ and θ = τ. These are alternative we w representations o transportation costs, and both lie between zero and one, with a higher value corresponding to a more open economy. Utilizing the act that all goods produced in a particular region have the same price, and using (8) or the number o varieties, we get Pw = H + ρw Pe = ρh + w P = θh + ρθw e ( H) + θw H e ( H) + ρθw H e ( H) + w H : Substituting these into the regional demand unctions, the equilibrium conditions or varieties produced in the three regions can be written as ρw e Pw θw Pw H Pw H + w Pe H + ρw e ( H) + θw H Pe e ( H) + ρθw w e ( H) Pe + ρθw e P P w H + w P H = = = Applying Walras Law, only two o the conditions are independent, and one nominal variable can be reely set. In what ollows I normalize the wage rate in the West to unity, w w =. Ater some straightorward algebra, the equilibrium conditions reduce to the ollowing Multiply the irst equation by ρw and subtract it rom the second, this leads to (9). Then multiply the irst equation by θv and subtract it rom the third, which leads to (0). 5
two equations: H + θw H = w e w e H + ρw e ( H) = w w ρw e ( ρ H) (9) θw H : θ (0) The equilibrium wage rates w e and w are thereore the solutions to (9) and (0). Proposition show describes the properties o the equilibrium solution. Proposition For a given regional distribution o human capital, the equilibrium wage rates w e, w are unique. Proo. See Appendix 4. It is possible to show 2 that w e P w =P e takes a very simple orm: wp W =P E = w 2 : Thus the migration equilibrium condition can be restated as h[ w 2 ( h)] D and h : () The inal step is to derive the equilibrium distributions o human capital across the West and the East: H = Z h 0 hdg(h) (2) The ull equilibrium o the model is described by equations (9), (0), () and (2). The our conditions deine the equilibrium values o w e, w, h and H. The migration condition holds with complementary slackness, which means that the initial symmetric spatial distribution o human capital in Home may be an equilibrium outcome i migration gains are small enough. 3 The impact o international trade In this section, I irst describe the closed economy outcome. Then I examine the impact o trade opening. Finally, I evaluate the quantitative perormance o the model by parameterizing the distribution unction G(h). 3. The closed economy Proposition 2 shows that initial symmetry is an equilibrium when Home is closed: Proposition 2 When the East and West have the same amount o human capital, there are no incentives to migrate. Thus spatial symmetry is a locally stable equilibrium. Proo. Using (9) with the assumption that θ = 0 (which corresponds to τ =, or that international trade is prohibitively costly), it is easy to see that H = =2 implies w =. This means that at a symmetric starting point marginal gains rom migration are zero (see []). As long as 6
Figure 2: The migration gain unction in a closed economy the migration cost D is strictly positive, there are no individual incentives to migrate even or the most skilled. It is possible to show that i the migration cost D is low, there is also an equilibrium with migration. To see this, note that the relative wage at the East, w, is a decreasing unction o H. Thus when people move rom the East to the West, the relative Western real wage increases due to scale eects. I migration costs are low, there is a level o migration that leads to migration gains that are greater than the costs (see [] again). The reason or multiple equilibria is that agents ace a coordination problem: only a sizable migration low leads to siginicant gains rom moving. Even in this case, however, migration does not lead to the complete depopulation o the East, because very low-skilled households will not move. Figure 2 illustrates the closed economy case. The curve B( h) h[ w 2 ( h)] is the migration gain unction or the marginal migrant (the let-hand side o []). There are three equilibria, o which two are stable: the initial situation and the equilibrium with migration on the increasing portion o B( h). The igure assumes that migration costs are suiciently low. I not, the only equilibrium is the symmetric one. 3.2 Trade opening Figure 3: The migration gain unction in the open economy Trading with Foreign breaks the symmetry o a closed economy. Since the West is closer to Foreign, it has a location advantage which leads to a higher wage rate (in the initial symmetric outcome). As a consequence, low enough migration costs make migration inevitable. Proposition 3 Trade opening leads to migration rom the East to the West, as long as migration costs are not too high. Figure 3 illustrates the impact o trade opening on a small economy. The upper panel shows the aggregate migration gain unction in autarky (broken line) and ater opening (solid line). 3 It is clear that or such a country international trade makes migration or the very skilled quite attractive. Also, it is interesting to note that - assuming a reasonable distribution or skills - migration gains drop substantially once the most skilled have moved. The middle panel shows how the relative real rental rate or human capital at the East changes with opening and migration. The impact o opening is a jump rom the broken line to the solid line at the right-hand end o the curves. Migration represents a movement along the solid line. While trade has a clear impact, quantitatively the eect is not very large. As a result o opening, (w=p E )=P W alls to 0.879. Small migration increases the dierence, but only 2 Substitute the right-hand sides o (9) and (0) into the price indexes. 3 The ollowing assumptions were used in the simulation. The skill distribution is set to β(2;4), which resembles the log-normal distribution, except that it is bounded on the interval (0; ). The other parameters used were = 4, τ = :2, µ = :5 and H F = 5. 7
modestly. For example, i 5% o the Eastern population migrates to the West, the relative real price o human capital urther alls to 0.864. What we observe in the statistics, however, is real income per capita. Thus we have to correct or the act that ater migration average human capital increases substantially in the West and decreases in the East. The lower panel shows the impact o migration on the Eastern relative real income. Given the 5% migration benchmark, GDP per capita alls to 0.767. I 0% o the Eastern population migrates, the numbers are 0.85 or the rental rate and 0.69 or real income. The overall impact o migration is the result o three separate eects. First, opening beneits the West because o its better location. This location eect is important, but cannot explain the emergence o large dierences alone. Second, migration changes the relative size o the regions, which in this model leads to an unambiguous increase in the Western rental rate o human capital. The scale eect, however, is small - mostly because the assumed distance between the East and the West is small. Thus modest migration alone contributes little to the wage change. Third, migration changes the skill composition o the two regions. The composition eect is sizable, similar in magnitude to the location eect. Moreover, as the igure shows, its impact is relatively more important at small migration lows. We can compare these results to the ones we would get without heterogeneous labor. In that case a 5% migration low would simply mean that the human capital level o the West increases to 0.525. Using (9) and (0) it is easy to calculate that the Eastern relative real wage alls to 0.872 (recall that it would be 0.879 without migration). The decline is smaller than with heterogeneous labor, but the dierence is not dramatic. In this case, however, relative per capita real income is also 0.872. Thus the composition eect lowers average incomes by more than 0%, and the dierence is even more dramatic at somewhat higher migration levels. To summarize, the model predicts signiicant income inequalities even within a small country when it opens up to international trade. Moreover, skilled migration is an important contributor to this inequality. In the simulation above, it explains almost hal o the total decline o the relative situation o the East. 4 Conclusion The paper presented a model in which international trade can cause substantial regional dierences in a country, even i there were none beore trade liberalization. The main driving orces are the possibility o migration, transportation costs, increasing returns and the heterogeneity o the population with respect to human capital. While the irst three elements are conventional in models o the New Economic Geography, heterogeneity (to the best o my knowledge) is a novel eature. The main conclusion o the model is that migration is a powerul ampliying orce o regional inequalities, i it involves the most skilled. Regional inequalities can emerge within a closed country in some cases as sel-ulilling expectations, and they are inevitable in a small open economy. As a result, average incomes diverge sharply, even i migration lows are small, because o the human capital reallocation (composition) eect. Finally, undamentals matter more than history in a small open economy with a mobile population, as the geographically disadvantaged region cannot maintain its earlier agglomeration advantage ater trade opening. The model delivers implications that can be tested empirically. Probably the most important implication is that the agglomerating region - the West in this paper - will have an income distribution that has a atter right tail than the depopulating region. This is because agglomeration 8
is driven by skilled migration, so migrants will add to the most skilled part o the workorce in the West. I much o ability is unobservable and only partially correlated with education, this phenomenon could be observed at any education/experience level. Another implication is that small actor price dierences might be compatible with large regional income inequalities. In particular, skill prices need not be very dierent in order to induce highly skilled people to migrate. Moreover, a small low o such individuals might not change skill prices very much, but per capita regional incomes will show big dierences. I skills are, as it is usually assumed, not completely observable, the composition and scale eects might be hard to disentangle. A shortcoming o the model is that it does not have explicit dynamics. In particular, the migration equilibrium might not represent a steady state, since uture generations can also choose to migrate. An easy way to remedy this is to assume that uture generations perectly replicate their parents skill distribution. In this case no urther migration would take place. Nevertheless, incorporating more general dynamics could lead to urther useul insights. While the model presented in this paper is relatively simple, I believe it successully highlights the importance o skilled migration in the generation o regional inequalities. Although urther research is clearly desirable, this paper can give useul suggestions to policymakers who want to understand the causes o regional income disparities. Reerences Davis, D. & Weinstein, D. (2002). Bones, bombs, and break points: The geography o economic activity, American Economic Review 92(5): 269 289. Dixit, A. K. & Stiglitz, J. E. (977). Monopolistic competition and optimum product diversity, American Economic Review 67: 297 308. Fujita, M., Krugman, P. & Venables, A. J. (999). Cambridge MA. The Spatial Economy, The MIT Press, Gallup, J. L., Sachs, J. D. & Mellinger, A. D. (998). Geography and economic development, Working Paper 6849, NBER. Krugman, P. (99). Increasing returns and economic geography, Journal o Political Economy 99: 483 499. Krugman, P. & Venables, A. J. (995). Globalization and the inequality o nations, The Quarterly Journal o Economics 0(4): 857 880. Matsuyama, K. (998). Geography o the world economy. Northwestern University. A The proo o proposition. Let us rewrite the equations that deine the equilibrium wage rates rom (9) and (0): A(w ;w e ) H + θw w e H B(w ;w e ) H + ρw e ( H) w e w w ρw e ( H) = 0 ρ θw H = 0: θ 9
Notice that in order to have a positive solution or the two wage rates, it is necessary that w e 2 ρ = ;ρ = and w 2 θ = ;θ =. The two equations implicitly deine two relationships between w e and w, and their intersection determines w e and w. There is a unique and stable (in the tâtonnemant sense) equilibrium i A w A we < B w B we () A w B w < A w e B we : To prove that this holds, it is suicient to show that A w < B w and A we > B we. 4 I will only derive the irst o these results, since the second one can be shown completely analogously. i H h θ 2 + ( ) 2 + 2θ 2 2θw θw θ 3 w B w + A w = 2 : w θ Let Q = 2 + 2θ 2 2θw θw θ 3 w : I this expression is positive, then B w + A w > 0. A suicient condition or Q > 0 is that min w Q > 0. Using standard calculus, it is easy to show that min w Q = 2 + θ 2 θ Thus the equilibrium wage rates are unique. s! 2 + θ 2 > 0: B Comparative statics with three regions Let = A w B we B w A we, which was shown to be negative in the previous section. Then the eect o H on w e is given by: dw e dh = (B w A H A w B H ) = B w + w e ρw e w e ρ < A w 2 ρw e + w e < 0; w e A w ( ρw e ) ρw e ρ where the irst inequality ollows rom B w > A w ;and the second inequality comes rom 2 ρw e > 2 ρρ ( )= = 2 ρ = > 0. 4 Note that A w < 0, B w > 0,A we > 0 and B we < 0. 0
To derive the eect o θ on w e, we can write dw e dθ = (B w A θ A w B θ ) " = < A w H B w w w H A w w w θ + θw w! θ w + θw!h # < 0: