An Economic Geography Model with Natural Resources

Transcription

1 An Economic Geography Model with Natural Resources María Pilar Martínez-García y Jose Rodolfo Morales z Abstract In this paper we present a dynamic model of trade between two regions which are endowed with natural resources (used for consumption and/or production). It is an economic geography model because it is based on the interaction of economies of scale with transportation costs as in Krugman (99). With this model, we show the existence of a "resource e ect". When the migration of workers between countries is not allowed, a shock of natural resources, brings down wages, for same scenarios, and reduces growth rates in the transitional dynamics. With migration, economies must grow at the same rate, but the one with greater endowment of natural resources, have a lower levels of industrialization and nominal wages. JEL classi cation: F2, F43, O4, Q0 Introduction The model is a variant in the monopolistic competition framework initially proposed by Dixit and Stiglitz (977). It is a dynamic version of the two-region model by Krugman (99) which incorporates di erences between the natural endowments of two regions with the aim of studying the causes of a resource curse. It can also be seen as an open-economy version of the growth model by Grossman and Helpman (99) which incorporates natural resources as consumption good. The main elements of the new economic geography like transportation costs and economies of scale are also present. The resource curse refers the paradox that countries and regions with an abundance of natural resources tend to have less economic growth and worse development outcomes than countries with fewer natural resources (Sachs and Warner, 997, 200). This paper analyzes the curse of natural resources using the approach of the New Economic Geography, which was initiated by P. The rst author acknowledges the support of MEC under project ECO This project is co- nanced by FEDER funds. Support is also acknowledged of COST Action IS04 "The EU in the new economic complex geography: models, tools and policy evaluation". y University of Murcia (3000). Facultad de Economía y Empresa, Room C409. Tel: pilarmg@um.es z University of Murcia. Murcia (3000), Alamo Street, N o 2. Tel: joserodolfo.morales@um.es (for any information, contact with this author)

2 Krugman in the early 990 s. The Economic Geography studies the causes of the uneven geographical distribution of economic activities and its evolution through time. Although there is abundant literature, theoretical and empirical, on the resource curse, the issue has not been addressed from the spatial perspective, as we do in this paper. In that context, imperfect market competition and migratory ows (of labor and rms) can a ect the economic growth rate, and the distribution of the industrial activity. We study the e ects of an unexpected increase in the natural resource sector in one of the countries. This is what happened in Venezuela with the Mene Grande discoveries in 94; in Nigeria with the discoveries of oil in the Niger Delta (956) and during the seventies in Norway, when the magnitude of its oil elds was discovered; among others. In all these examples, the resource curse hypothesis was satis ed. The countries experimented a decrease in the growth rates. The model without migration, predicts that a positive shock in the resource endowment in one of the countries, will rise the wages in this country in the short-run; and lower their growth rates in the transition, back to the sustainable balance growth path, in the long-run. When migration ows are allow, both economies must grow at the same rate, along a balance growth path, to ensure the equality of real wages. However, the distribution of the industry presents a negative relation with the natural resource endowments of the regions, as a consequence of what we call: the resource e ect. The paper is organized as follows: Section 2 presents the model; Section 3 is devoted to the study of short-run; Sections 4 and 5 are dedicated to the long-run without migration ows; Sections 6 and 7 discuss the case with migration; and section 8 concludes. 2 A dynamic two-region model We consider a model of two regions. In this model there are assumed to be two kinds of goods: manufactures, which are tradable goods produced by an increasing-returns sector that can be located in either region, and natural resource, which is considered as an endowment that each region possesses. These endowments have the characteristic of being non-tradable across regions. The number of varieties of industrial goods n and n 2 for each region can be enlarged devoting e orts to the R&D sectors. Trade is not free, their exists transportation cost. The production of industrial goods needs labor and natural resources. Moreover, there are xed costs that guarantee the existence of economies of scale. All individuals in this economy are assumed to share the intertemporal utility function of the form Z h i U ln C M C e t dt; () 0 A 2

3 where C A is consumption of the natural (agricultural) good and C M is consumption of manufactures aggregate which is de ned by Z n C M ; (2) 0 Z c n2 i di + 0 c 2i being n and n 2 the large number of potential products from each region at a given instant of time and > is the elasticity of substitution among the products. Individuals hold also assets in the form of ownership claims on innovative rms. We denote the net assets per person in region j by a j (t). Individuals are competitive in that each takes as given the interest rate, r j (t), the price of the natural resource p Aj (t), and the wage rate, w j (t), paid per unit of labor services. Each individual supplies inelastically one unit of labor services per unit of time. The total income received by an individual per unit of time is, therefore, the sum of labor income, w j, and assets income, r j a j. Individuals use the income that they do not consume to accumulate more assets: Xn _a j w j + r j a j p Aj C Aj i Xn 2 p i c i i p 2i c 2i (3) where p ji is the price of the good i produced in region j and < is the constant transportation cost. Transportation costs for manufactured goods will be assumed to take Samuelson s "iceberg" form, in which transport costs are incurred in the good transported. And nally, L j is the total labor force in region j. The individual s optimization problem is to decide on variables C Aj, c i and c 2i to maximize U in equation () subject to the budget constrain in equation (3). The usual Euler equation arises, for both countries, that is _E E r ; _E 2 E 2 r 2 (4) where E j denotes expenditures in consumption (manufactures and natural resource) in region j. And the household demand for primary goods is: C Aj ( ) E j p Aj (5) We can now turn to the behavior of rms. The production of an individual manufactured good i involves a xed cost and a constant marginal cost, giving rise to economies of scale: x i l x i j A i where l xi is labor used in producing i and x i is the good s output. The xed cost j depends on the region j and it is de ned as j n j, that is, the higher (6) 3

4 the number of rms in a region n j the lower the xed cost. The production function (6) is similar to the one proposed by Krugman (980, 99). We assume that there are a large number of manufacturing rms, each producing a single product. Then, given the de nition of manufacturing aggregate (2) and the assumption of iceberg transportation costs, the elasticity of demand facing any individual rm is (see Krugman 980, 99). The pro t-maximizing pricing behavior of a representative rm in region is therefore to set a price equal to p w pa (7) and the resource demand equal to: A w l xi p j A where w is the wage rate of workers in region. Since rms are identical and they face the same prices, labor force l xi l for all i. x w p A l xi j Similar equations applies in region 2. Comparing the prices of representative products, we have p w pa (9) p 2 w 2 At a point in time, the technology exists to produce n (in country ) and n 2 (in country 2) varieties of consumption goods. An expansion of these numbers requires a technological advance in the sense of an invention that permits the production of the new kind of consumption goods. Following Romer/Crossman- Helpman/Aghion-Howit, the change in n j (j ; 2) will be equal to the number of people attempting to discover new ideas, l nj, multiplied by the rate at which R&D generates new ideas, j : p A2 (8) _n j j l nj (0) where j n j in its simplest form. Any individual is allowed to enter the R&D sector and prospect for new designs, so that labor must receive the same compensation in its two uses: w j j V j () where V j is the net value of future expected pro ts linked to the discovery in country j, whose expression is Z V j (t) j (s)e rj(t;s)(s t) ds with r j (t; s) r j (v)dv (2) t (s t) t j (s) p j x j w j l xj p Aj A j (3) Z s 4

5 Function j (s) represents instantaneous pro ts of monopolist. Di erentiating (2) it is obtained that r j (t) j V j + _ V j V j ; (4) that is, the rate of returns r j equals the rate of return to investing in R&D. The R&D rate of return equals the pro t rate, j V j, plus the rate of capital gain or loss derived from the change in the value of the research rm, _ Vj V j. Combining equations (7, 6, and 3), it yields that j (t) ( ) V j(t) L Ej [ + ( )] (5) where L Ej P n j i l x i : While in the natural resource sector, a typical primary rm will maximize their bene ts, choosing the amount of labor to employ in the extraction of the resource, subject to the extraction function: max j p Aj j w j l j s:t : j S j l j (6) The extraction, yield or harvesting (we use these terms to refer also to the extraction, as is done in the bio-economic literature) of the resource is a direct function of: the available stock of the natural resource in the region, S j, the labor employ in the extraction, l j, and j, that is a productivity parameter. The rst order condition of this maximization problem is: p Aj j l j w j (7) Notice that this condition also implies that the pro ts of these rms will be equal to zero. 3 Short-Run Equilibrium As we study the short term, in this section we assume that the population of each region is given. Let c be the consumption in region of a representative region product, and c 2 be the consumption in region of a representative region 2 product. From the rst order optimality conditions derived from the optimization of () subject to (3) it is obtained that c c 2 p p 2 If z is the ratio of region spending on local manufactures to that on manufactures from the other region, and z 2 is the ratio of region 2 expenditure on region products to spending on local products, we have: 5

6 z n c p n 2 c 2 p 2 (8) z 2 n c p n 2 c 2 p 2 (9) The expenditure in manufactured goods produced in one region, have to pay all the factor involved in their production, plus the pro ts, that is: z z 2 n + p A A n + w L E E + q E 2 + z + z 2 E 2 n 2 + p A2 A 2 n 2 + w 2 L E2 + z q + E 2 + z 2 (20) (2) The terms on the right, of expression (20) are the region and 2, expenditures, devoted to industrial goods produced in region. While, on the left are the payments to productive factors, plus the bene t of the industrial rms. Where q are the terms of trade; equal to the ratio of wages (q w w 2 ). Expression (2), states the same equality for region 2. These expressions are very similar to those of Krugman (99), di ering in that, in our model, work is not the only factor of production, and spending it is not necessarily equal to income. Making use of (8 and 3) the previous equations can be expressed as follows: n + w 2 n 2 w 2 + z " + z 2 " 2 + z + z 2 " + " 2 + z + z 2 (22) (23) where: " j Ej w j The above equations represent the equilibrium in the market for industrial goods. Now that we already have an expression to determine the bene ts, equation (3) gives the amount of labor in the industrial sector: j n j L Ej ( ) + + (24) In the equilibrium, the supply of natural resource, equation (7) must be equal to the sum of individual consumption (5) and industrial use (8) in each country, which gives: j n j l j ( )" j + ( ) ( ) + (25) Turning now to the R&D sector, the total investment of a region must be equal to the total income minus the expenditure, in that region. Taking into w j w j 6

7 account that (A j n j V j ), and using expressions (0), () and (4), the labor engaged in research can be obtain: l nj L j + jn j w j " j (26) We also assume full employment, so that total labor force must be exhausted by labor used in production, in R&D sector and in the resource sector: n j X L j l xi + l nj + l j L Ej + l nj + l j (27) i Then, the markets equilibria can be summarized by the system (22) - (27). Nevertheless, we can do some manipulations of this equations in order to simplify the system. Firstly, replacing all the labor forces, into the full employment condition, we get the equilibrium in the labor market; if and only if: j n j w j + " j (28) Secondly, with this equality the system can be rewritten as: where: X n w n 2 w 2 " X T + X (29) " 2 + XT ( ) L Ej " j + (30) [ + ( )] l j " j (3) l nj L j " j (32) S S 2 ( )( ) and T : We turn now to study the consequences of di erent shocks, over the wages, in the short-run. The following propositions study the e ect of a suddenly increase of the natural resource, the variety of industrial goods, and the expenditures, in one of the regions: Note that, when a shock in the natural resources take place, the distribution of the labor forces between the di erent sectors won t change. This happen because, from the beginning, we are assuming that wages, in each region, are equal across sectors. Another interesting fact is that, due to the xed cost, each region must have at least a minimum population, allowing industrial production and resource extraction, but without people dedicated to research. In a case like this, the region only produce a nite and constant amount of variety of industrial goods. As a reference, let s see what happens when both regions are equal. First, lets rewrite equation (29) as follows: (" " 2 ) T + (" " 2 ) X X 0 (33) 7

8 Making use of the de nition of X, the derivative of the last expression with respect to the rate of wages is: ( ) w w 2 (" " 2 ) X + X > 0 (34) In the situation of symmetry, we have that L L 2, " " 2, n n 2, S S 2 and 2, this implies that the labor devoted to each sector is the same in both regions (l n l n2, L E L E2 and l l 2 ). Is easy to see that w w 2 is a solution in this case, and considering (34), this is a unique solution. Now we can get some more questions related to the curse of natural resources, although in the short term. Such as: What happens when new sources of natural resources are discovered, or when the state of the environment facilitates its regeneration? Propositions and 2 shed some light on this: Proposition Starting from a position of symmetry between the two countries, when the natural resource is devoted only to nal consumption ( ), a shock of natural resources that takes place in one of the two regions, such that S S 2 >, will not modify the equilibrium wage rate, and hence w w 2. Proof. See the appendix. If a suddenly increase of S happens, since it is not a tradable good, the excess of natural resource supply in region will generate a decrease in the price of this resource until the population of the same country be willing to consume all of it, so their incomes will not be a ected by this shock. But if (0 < < ), changes in the price of the resource will a ect the price of industrial goods, and wages will change due to, what we have called, the resource e ect. To isolate this e ect, in the next proposition, we hold constant the expenditures and the ratio of variety of goods, allowing the ratio of resource stocks, to change. Proposition 2 When the natural resource is devoted to nal consumption as well as to industrial production, (0 < < ), an increase in the resource endowment of one of the regions, will increase the wage of this same region. Proof. See the appendix. Starting form a symmetric position between regions, if the ratio of resource endowments increases, due to an external shock, since the resource is not tradable, the ratio of prices (p A p A2 ) will fall, and initially, the aggregate expenditure will not be a ected (as in the case ). However, the industrial rms in region will bene t from a cheaper input. This will allow them to reduce the price of industrial goods, increasing sales at expense of rms in region 2. This causes an increase of the exports and a reduction of the imports of region. The trade balance of region 2 moves in the opposite direction. This increased demand for industrial goods in region, generates upward pressure over the price-ratio and wage-ratio, that will increase until the pressure disappears, i.e. when the trade balance nd again the equilibrium. 8

9 Proposition 3 An increase of the aggregate expenditure of one of the regions will reduce the wage of this same region, due to what is called the "expenditure e ect". Proof. See the appendix. There are two ways to explain this e ect: through the trade balance, or through the markets equilibria (goods and labor). Considering the trade balance, equation (76) of the appendix, what happen is that, at the initial wages, an increase of the aggregate expenditure (in region for example), will rise the imports of region, while export reduces. This causes an excess of supply of industrial goods produced in region, and an excess of demand of the goods produced in region 2, generating a downward pressure on the price ratio (p p 2 ). Then, wages (w w 2 ) and prices must fall, in order to reduce imports and to increase exports, on industrial goods from the region, until the equilibrium is restored. If we focus on one of the regions, Figure illustrates the perspective of the markets equilibria. L w represents the labor market equilibria, equation (28), and the G w represents the industrial goods equilibria, equation (22); as functions of the ratio of wages. Real Profits L'(w) L(w) G(w) G'(w) wages Figure : Expenditure and Variety E ects where: Real pro ts j w j + j When expenditure increases, in region for example, both demands, for industrial goods and for labor, are a ected. In the goods market, new spending is used to buy industrial goods from both regions, therefore, the demand for local goods will increase, but by lower proportion than spending. In the labor market, demand will decrease, due to the contraction in the research sector. Since, income is xed and spending is now higher, the investment is lower; so that fewer researchers are engaged in the R&D sector. The economy moves from it s initial position to the point where red line G 0 w crosses the initial wages. There is an excess of labor supply, because real pro ts 9

10 are too low, and rms are not willing to continue hiring labor. This imbalance in the labor market, creates a downward pressure on wages, which will be reduced gradually, driving up the real bene t of the companies, until they absorb all the labor. The result is an unambiguous reduction in nominal wages. But the sectorial composition is di erent too. R&D sector shrank due to increased spending. And although the rise in pro ts cushioned this fall, part of the workforce shifted to the industrial sector and to the primary sector. So the investment is now lower than before, and the production of consumptions goods is higher. This expenditure e ect works in the same direction as Krugman s competition e ect but, the economic intuition behind both, is di erent. The competition e ect occurs because a larger number of industrial rms have to compete for the same market. In our model, the competition e ect is not present due to the non-tradable primary good. This is specially evident once we consider Proposition (), where the endowments of resources does not a ect the incomes. While in Krugman s model, the tradable agricultural good, have a direct impact over the incomes of the regions. The following proposition explains a second e ect, that we call: variety e ect. It takes into account the creation of new varieties of industrial goods, assuming that the regions are symmetrical in their expenditures and resource stocks. Proposition 4 When a region produces a greater variety of industrial goods, the wage of that region will tend to rise, due to the "variety e ect". Proof. See the appendix. According to the trade balance equation (76); since consumers prefer variety to quantity, the proportion of expenditure, form region and 2, dedicated to industrial goods of region, will increase if n > n 2. At the initial prices, this will causes an excess of demand of goods produced in region. Then, prices and wages must increases too. According to the market equilibria; now the initial equilibrium is the intersection of G 0 w and L 0 w in Figure. The demand for each of the industrial goods will fall, because now there is a larger variety of goods to consume. The decline in demand for each variety of industrial good, will be divided between the varieties produced in region and region 2. On the labor market, the aggregate labor demand increases, because now there are more rms in the economy. Only for analytical purposes, assume the economy moves from it s initial equilibrium to the intersection between the initial wages and G w ; where there is an excess of labor demand. Individual pro ts are still high, so more rms have incentives to continue demanding labor. The pressure over the pro ts and wages makes the economy to move along the curve G w until the bene ts are su ciently low and wages high enough that companies demands the right quantity of labor that exist in the region. As a result, the wages are now higher, and the individual pro ts lower. Nevertheless, the aggregate pro ts,, doesn t change, therefore, the productive structure of the region, that is, the size of the sectors, remains the same. 0

11 As we are talking about economic geography, one might ask: How does this variety e ect is related with Krugman s home market e ect? Indeed they are almost the same thing. To arrive to Krugman s home market, in our model, we should keep constant the ratio of spending per company (" j n j ). But, in the variety e ect, while increasing the ratio of variety of goods (n n 2 ), the spending per company falls. This means that labor demand per industrial rm increases. The imbalance in the labor market, creates pressure to raise wages, reinforcing the home market e ect. Before turning to analyze the long-run equilibrium, we have one more question: What happen with transportation costs? The following proposition try to shed some light on this issue. Proposition 5 a) If expenditure of region is greater than the expenditure of region 2 (" " 2 > ), an increase of will reduce the ratio of nominal wages. b) When region 2 has the biggest expenditure (" " 2 < ), an increase of will rise the wage ratio. c) And, if both regions are equal (" " 2 ), an increase of won t change the ratio of nominal wages. Proof. See the appendix. The e ect of an increase of depends on the size of the economies. When the transport cost reduces, the locals will transfer expenditure from local to foreign goods, if we are in case a), where the region is larger, in terms of total expenditure, the households in this region will nd that import industrial goods from region 2, is now cheaper, so imports will grow. At the same time, industrial goods of region will become more accessible for residents of region 2, so exports of region will grow too. Since the aggregate expenditure of region is greater, imports grow more than exports, resulting in a trade imbalance, at the initial prices. The only way to correct the excess supply, is reducing prices, i.e., reducing the ratio of wages. The opposite happens when the economies are in case b); the exports grow more than the imports, because the foreign market is "bigger", in terms of expenditures, generating an excess of demand, pressing prices and wages upward. When both economies have the same size, case c), the reduction in the transportation costs, causes displacement of local spending by foreign spending in equal measure. Exports and imports increase by the same amount. This is a clear implication of the new economic geography. The reduction in the transport costs weakens the home market e ect. The barriers to international trade are lower, then it becomes less relevant where companies are located. In conclusion, the largest economy, will su er in terms of nominal wages due to lower transport costs, while the small economy will be bene t. 4 Long-Run Equilibrium without Migration Now that we have established the behavior of the economies in the short-run, we can begin to study their evolution over time. However, we need more information

12 about the dynamic of the resource stocks. Regarding this, we can assume that its dynamics are driven by a logistic function: _S j g j S j S j l j g j S j j (35) CC j CC j (36) We are considering renewable resources, which reproductions depends on the intrinsic growth rate of the resource (g j ), the stocks of resources at each moment of time (S j(t) ), and the carrying capacity or saturation level (CC j ), minus the extraction or harvesting; equation (35) is asymptotically stable. The constant sustainable stock is determine by equation (36); where l j can be interpreted as the e ort in the extraction of the resource. Then, the sustainable stock, in the long-run, is a decreasing function of the e ort. We are interested in nding sustainable solution, because it is the only one that can be maintained in the long term. Then, to ensure that: Sj > 0, it must: g j < l j. Replacing Sj into the extraction function (6), we obtain the yield-e ort curve, or Shaefer Model, Figure (2): H* g/2 g effort Figure 2: Yield-E ort curve where H is the harvesting in the long-run. The maximum sustainable yield (MSY) is reached when l j g j 2. Before that point, any extra e ort leads to an increase in the extraction, and beyond that, the extraction will decrease. It is worth emphasizing that the yield-e ort curve represents only the equilibrium yield (sustainable extraction) j corresponding to di erent levels of e ort. Therefore, it should not be interpreted as an increased e ort beyond MSY will lead to an immediate reduction in the S 0 can also occur in the long-run, but it is a solution in which the economy does not produce any good, and therefore is not an interesting solution for analysis. 2

13 extraction. Equation (6) implies that the short term extraction will always increase with the e ort; it is only over the long term, when the processes of resources dynamics have resulted in a decreased resource stock, that yield ultimately declines. To this point, we can thus summarize the evolution of the economies by a system of four di erential equations in the variables: ", " 2, S and S 2. Substituting equation (4) in the euler equations (4); taking into account the R&D production function (0), the free entry condition (), and equation (26), we have: _" " [ (" L ) ] (37) _" 2 " 2 [ (" 2 L 2 ) ] (38) S [ + ( )] _S S g " (39) CC S 2 [ + ( )] _S 2 S 2 g 2 " 2 (40) CC 2 Using the de nition of the variable X and equation (29), we arrive to the growth rate of the wage ratio: w ( ) S + n + XT " "2 + X 2 (4) (" 2 " ) where w w w2, n n n2 and S S S2. In this section of the paper, we continue to assume that there is no migration between the regions. In a perfect-foresight growth equilibrium, each agent behaves optimally (taken as given the time paths of variables out of their control), and the growth rates of all variables are constant (", " 2, S and S 2 ). A quick inspection of equation (29), reveals that X _ 0 when both economies grow at the same rate. From now on, let j (j ; 2) be the growth rate of the economy, in the BGP; and asterisk in the superscript of a variable, indicates the equilibrium value. The solution of system (37) - (40) is: " L + (42) " 2 L 2 + S g S 2 [ + ( )] [ + ( )] g 2 (43) L + CC (44) g L 2 + CC2 (45) g 2 The distribution of the labor force (L L 2 ) is given, because of the nomigration assumption. And the resources stocks (44) and (45) ensure a sustainable path for both economies. By using the BGP values of the variables, and the system (29) - (32) we arrive to the following proposition: 3

14 Proposition 6 Given a population distribution (L L 2 ), and some initial conditions for the variety of goods and the resource stocks, a unique positive solution for the system (29) - (32) and (42) - (45) exists, along the balance growth path: q X (" " 2 ) T + 2 (" " 2 2 ) 2 T (" " 2 ) L E j ( ) (L j + ) + (46) l j [ + ( )] (L j + ) (47) l n j (L j + ) ( + ) (48) Proof. See in the appendix. In the BGP, the economies growth rates are: j w j. We can only nd the di erential growth rates between the two economies ( 2), just what equation (4) shows, but with the rst and third terms equal to cero: n (49) ( ) (L L 2 ) (50) Proposition 7 In a BGP, the ratio of aggregate consumption expenditure (E E 2 ), grow at the rate given by (50). Proof. This comes straightfoward from equation (4) and the de nitions of " and " 2. Since rj > j, household utility is always nite, and the relevant transversality condition is satis ed. This is the standar result in the endogenous growth models, where the growth rates depends on the population sizes. A few things can be said of distribution of the labor force between sectors and the growth rate, through equations (46), (47), (48) and (49). When the economies of scale increases (a reduce of ), the industrial sector is more e cient and the labor force, released by this sector, can be devoted to research activities; although both regions will increase their work in research, it increases more in the most populous country, accelerating it s growth. If what rises is the proportion of the expenditure in industrial goods (), the demand for this goods will be greater, and more labor will be devoted to production and research of new varieties of goods; the primary sector will decline, and the largest region will bene t again, in terms of growth rates. If () is bigger, the rate of success of the research is greater, other things equal, the R&D sector is more productive, attracting labor force from the others sectors, promoting the growth. Thus, when the productive structure favors the industrial sector (low value of and high value of ) and innovation (high value of ), the region with the largest populations can gain a greater advantage, in terms of growth. 4

15 5 Stability and Transition without Migration In this section, we analyze the stability of the unique BGP found in the previous section. That is, although the economies might eventually grow inde nitely, could this sustained growth path be reached if the economies are not initially in equilibrium? The eigenvalues associated with our di erential system, equations (37) to (40), are: e L + > 0 (5) e 2 L 2 + > 0 (52) [ + ( )] e 3 g L + < 0 (53) [ + ( )] e 2 g 2 2 L 2 + < 0 (54) Where the model has two state variables (S and S 2 ), therefore, the BGP can be characterized as a saddle-point equilibrium. 5. The e ects of a resource shock on short and long-run equilibrium We have now all the elements to study the consequences of a resource shock on growth. It is obvious from equation (50), that a shock in the endowment of natural resources, does not a ect growth in the balance growth path. The longrun growth depends exclusively on the size of the populations. This is always so in R&D based endowenous growth models. However, it does have e ects on wage levels, and growth during the transition to the long-run equilibrium. Because we are interested in the resource curse; assume that economies are in their respective balanced growth path, and that at some point in time, a positive shock take place, such that region have, a resource stock greater than the equilibrium level (S > S). This shock can be the result of an increase in the productivity of extraction. We name 0 j to the increased productivity, and S to the new sustainable resource stock. The e ects over the transition to the balance growth path, are complex, and relies on: the populations sizes, the magnitude of the resource shock and the values of the parameters. To understand the dynamics triggered by the shock we have to know three things: ) where are the economies just before the shock, i.e. what happens in the short-run; 2) which is the new long-run equilibrium; and 3) how does the economies reach it. So, basically: where are we, where are we going, and how we get there. We have already answer the rst issue when we discussed the resource e ect. Although, Proposition (2) refers to an increase in the ratio of resource stock, a shock over the ratio of productivity 2, have exactly the same e ects, leading to a higher wage ratio. 5

16 Regarding the second issue, the new long-run equilibrium could be above the initial equilibrium, which implies a higher ratio of wages; or below, which implies a lower ratio of wages. What determines this, is the yield-e ort curve, Figure. The increase in the productivity of the harvesting, could be seen as a rise in the e ort. If the e ort is lower than g j 2 the extraction in the long-run will reduces; the resource supply in the long-run will be lower, and the resource price higher. Then, the industrial price ratio, and the wage ratio, will be higher than the initial values. Exactly the opposite occurs when the shock leads to an e ort beyond the MSY. One more thing is that, even when the new long-run wage ratio is higher, it will be lower than the wage ratio of the short-run. And, why is that? Because, the e ort shock, in the short-run, makes p A p A2 to rise above it s initial level, but in the long-run, when the resource dynamics begin to take place, the resource stock diminishes. So, even when increases, this equilibrium extraction will be lower than the of the short run. Finally, how the economies goes from the short-run equilibrium after the shock, towards the new long-run equilibrium? Lets call T to di erential growth rates in the transition: T ( ) S + n Equation (55) is equivalent to equation (4), when " and " 2 are constant. An implication that arises from this equation is that: populations size matters. The transitional growth rate will be negative when L 6 L 2, because S < 0 and n 6 0. However, if L > L 2 the transitional growth rate can be positive or negative, because S < 0 and n > 0. In this second case, is the size of the shock (4S t S t S ) what determines the sing of the transitional growth rate. When the shock is lower than certain threshold (4S th ), > T > 0. And, when shock is bigger, such that 4S t > 4S th, the transition have two faces: T < 0 < and > T > 0. Where the threshold is: 4S th (L L 2 ) CC g ( ) ( ) (55) (56) Summarizing, in the short-run, wage ratio increases; in the long-run, it depends on the position over the yield-e ort curve; and the transition growth rate depends on the magnitude of the shock. The following table summarizes this results: 6

17 L > L 2 w w 2 0 l < g 2 0 l > g 2 4S t < 4S th 4S t > 4S th A B C 0 < T < 0 < T < T < 0 < r pa p A2 L L 2 2 w w 2 r w w 2 pa p A2 T < 0 T < 0 r w w 2 r pa p A2 T < 0 < L < L 2 3 w w 2 r pa p A2 pa p A2 T < 0 < r w w 2 pa p A2 r w w 2 pa p A2 As the table shows, there are seven di erent cases. Lets start with the more "interesting" one, concerning the resource curse, case.c. Initially, region is growing faster than region 2. Just for analytical purpose, assume that both regions have the same productivity in the harvesting ( 2 ). A positive productivity shock occurs, such that 0 >. In the short-run, the resource e ect take place. The primary good is cheaper due to the increase in the supply, as we saw, a higher e ort always leads to a higher extraction in the short run, and the industrial rms take advantage of this situation. The exports of the region grow, and the excess of demand for industrial goods produced in region, start pushing prices upward; nally, the ratio of wages rise until the excess demand vanishes. This results in a discrete jump in the ratio of wages, as we see in Figure 3. However, as times goes, region harvesting start to diminish because the resource stock reduces. This gradual fall in the supply, makes the price ratio of the primary good to rise. Due to the high extraction that took place just after the shock, the stock of resources falls even below the initial stock (S); and, even more, the environmental "damage" is so great that the extraction is also reduced below its initial level ( ). So the price of the primary good rise above it s value before the shock; the primary good is now more expensive than before! Industrial rms from region, that require the primary good as an input, see their costs going up, as the harvesting contracts. The industrial price ratio starts to increase too, due to the expensive input. On the other hand, households replace industrial goods from the region, for goods produced in region 2, which are now comparatively cheaper. If we look at the balance of trade, equation (76), imports are growing and export diminishing, so it s not in equilibrium; or in other words, there is an excess of supply of industrial goods produced in region. The excess of supply generates a downward pressure over the industrial price ratio, so rms will pay less and less to their workers, until the resource reach it s sustainable level, and 7

18 the harvesting stabilize at his long-run value ( < ). w t Figure 3: Case.C The scenario we have just described, has all the elements of a resource curse. Brie y, region experience a jump in their nominal wage in the short-run, due to the shock in it s primary sector; then the resource dynamics triggers a negative transitional growth rate, after a while this rates turns positive, but lower than the long-run growth rate, until the point where the region converge to it s new equilibrium. However, the ratio of wages in each moment of time, is now lower than before the shock. As in the case of, certain regularities are observed with respect to proindustry and pro-primary economies. When the values of the parameters are such that favor the primary activity (high values of and ; and lower values of and ), the growth rate during the transition to the BGP will be lower than if the production structure favors the industrial sector (high values of, and ; and lower values of and ). This happens because the parameters a ect both, the size of the shock, and the threshold, in opposite directions, accentuating adverse e ects in the case of pro-primary economies. Furthermore, low values of accentuates the resource e ect, which decreases the ratio of wages in the long-run. What about the other cases? In case.b, Figure 5, the situation is very similar. The only di erence is that the shock is smaller than the previous case, so the wage doesn t need to decrease, simply slow down their growth, until the new equilibrium is reached. Here, the region doesn t have negative growth rates during the transition, but the rest is exactly the same. This scenario could be considered too as a resource curse. Case.A, Figure 4, is quite di erent however. The increase in the productivity does not imply a large rise in the harvesting e ort. So, during the transition, the supply of primary goods will decrease but, it will stabilize at a superior level than the initial extraction ( > ). Therefore, the dynamics will resemble the case.b, but the nal result will be the opposite. Here, the shock bene ts 8

19 the region. w w Figure 4: Case.A t Figure 5: Case.B t w w Figure 6: Case: 2A t Figure 7: Case: 2.BC t w w Figure 8: Case: 3A t Figure 9: Case 3.BC t Cases 2.A and 3.A are similar to.a, and the economic intuition behind is the same, where the shock bene ts the region. While cases 2.BC and 3.BC resembles to case.b or.c, where a resource curse appears due to the shock in the primary sector. 6 Migratory Flows Until this point, we had analyze a model where people work in the same region where they were "born". In this section, we want to change this assumption, allowing migratory ows between regions. We need to model this ows. First, we have to keep in mind that workers are interested not in nominal wages, but in real wages. Following Krugman (99), the real wages of workers in each region are:! w P p! 2 w 2 P 2 p ( ) A (57) ( ) A 2 (58) 9

20 with P j the price index of manufactured goods for consumers in region j, given by P n p p2 + n 2 ; (59) in region and, that for consumers residing in region 2 is P 2 n p + n2 p 2 : (60) Price indexes P j are weighted sums of the manufactures prices given in (7). Lets normalize the total population to one (L + L 2 ), where the dynamic of the population is driven by:! _L L ( L )! 2 On the long-run, the labor force will move between the two regions until both real wages will be equal. Note that when L approaches the unit, L _ slows down, until it becomes zero. The same happens when L approaches zero. From equations (9), (7) and (29), and after some manipulation: 82 >< _L L ( L ) 4 n S >: n 2 2 S 2 ( )( ) " " 2 X > >; (6) Where X is a function of " and " 2 through equation (29). Using equations (0) and (32) we found our last di erential equation: n ( ) _n (2L ) (" " 2 ) (62) n 2 where: n n n 2 Now, the dynamics of our economies can be summarize by the system (37), (38), (39), (40), (6) and (62), in the variables: ", " 2, S, S 2, L and (n n 2 ). 7 Long-Run Equilibrium with Migration In this extended model, the Balance Growth Path is more restrictive, implying that: _" _" 2 _ S _ S 2 _n _ L 0. For now, lets focus only on the interior solutions, therefore two conditions must hold:!! 2 and 0 < L <. Before beginning the analysis of the model itself, the following propositions shows the existence of an interior solution, and characterize it. 20

21 Proposition 8 Along the BGP there exist unique solution for L and L 2 for which real wages are equal in both economies. " 2 + (63) " (64) S [ + ( )] g 2 + CC (65) g S2 [ + ( )] g CC2 (66) g 2 n S S 2 ( )( ) (67) L 2 (68) Proof. See in the appendix. Observe that we have the "standard" result of the model without migration, for the variables ", " 2, S and S 2. As Proposition (7) states, the aggregate consumption expenditure in both regions, hence the economies, grows according to (49). However, the size of both economies L and L 2 are not given, they are determined by expression (6). In the sustainable BGP, the population is equally distributed between regions. Proposition 9 In the sustainable BGP, the industrial price indexes are equal (P P 2 ). Proof. See in the appendix. The regions having the same sizes imply that their expenditures, expressed in a common currency, are equal too (" w " 2 w 2 q). In this case, to maintain the trade balance in equilibrium, equation (76), the proportion of expenditure devoted to imports, must be equal to the share devoted for exports. In other words, the relative weight of the set of goods of region, against the set of goods in region 2, must be equal to one (X ). Ultimately, this means that both baskets of goods have the same weight. Because transportation cost are equal in both regions, and now, for the equilibrium of the trade balance, the baskets of industrial goods weight the same, the industrial price indexes are equal. However, the composition of the basket of goods (n j, w j and S j ), may vary among regions, depending on what are the initial conditions and the value of the parameters. Proposition 0 In the BGP, both economies grow at the same rate ( 0). Proof. It is obtained by substituting L into equation (49). This comes as a corollary of the condition that ensure a sustainable balance growth path, and an interior solution. In terms of growth rates, this condition can be write as: 2

22 w w2 P P2 + ( ) pa pa2 As a consequence of Proposition (9) and equation (29), the industrial price indexes are constant in the long-run equilibrium. According with the previous equation, the ratio of wages can only grow if the price ratio of the resources grow too. But, since we are in a sustainable equilibrium, the stock of natural resources are constant. To maintain the real wages equal (i.e. to be in the interior solution), nominal wages must remain constant over time. 7. E ects of Di erences in the Natural Resources Although, in the model with migration, economies grow at the same rate, we can analyze what happens to the level of industrialization, and nominal wages, when a region has more natural resources than the other. Proposition In a BGP, the economy with greater stock of natural resources will have lower wages and less variety of industrial goods, according to: w w 2 n n 2 S S 2 S S 2 ( ) ( )( ) Proof. See in the appendix. Suppose that region have a greater natural resource stock than region 2. The supply of primary goods is greater in the region with more natural resource stock, making its price lower. As P P2, according to Proposition (9), the nominal wage should be higher in the region with the lowest natural endowment to maintain the balance of real wages, and to avoid migratory ows. However, with other things equal, the highest nominal wage create a trade imbalance between regions. Spending on goods in region are now greater than the income of the region. Or, exports of region, exceed imports. To prevent this imbalance, region 2 must have a greater variety of industrial goods, to attract the excess demand, through the channel of the home-market e ect. Proposition 2 An increase of won t cause any changes. (69) (70) Proof. See in the appendix. Because both regions are equally populated, the relative weight of the set of goods from both regions are equal (X ), so the industrial price indexes are equal too, as we know from Proposition (9). In this context, a reduction of the transportation costs, bene ts the regions equally, maintaining P P 2 and!! 2. While imports and exports increases in the same amount in the two regions. As a result, nor the wages-ratio, nor varieties-ratio, changes. 22

23 8 Conclusions We have studied a model of trade between two countries. Both regions are endowed with a renewable natural resource which is used as input and it is also a nal consumption good. The natural resource is not tradable. The model has the main elements of the new economic geography like transportation costs and economies of scale. In the model without migration, a positive shock in the resource sector, causes lower growth rates, or even negative, during the transition to the longrun equilibrium, and a lower level of wages for some scenarios, due to the resource e ect. While, in the model that allow migratory ows, both economies grow at the same rate, but, there is a negative relationship between natural endowment and industrialization. This comes as a result of the interaction of the resource, expenditure and variety e ects, together with the real wage equalization. The next step in our research is to identify the transitional dynamics of the model with migration, in order to clarify the interactions, as we have done for the model without migration. 9 References Baldwin, E. R., Martin, P. (200) Global Income Divergence, Trade, and Industrialization: The Geography of Growth Take-O s, Journal of Economic Growth, 6: Dixit, A. and Stiglitz, J. (977) Monopolistic competition and optimum product diversity, A.E.E. 67, Grossman, B. Helpman, E. (99) Innovation and growth in the world economy, MIt Press. Cambridge. Krugman, P. (980) Scale economies, product di erentiation, and the pattern of trade, A.E.R, Krugman, P. (99) Increasing returns and economic geography, Journal of Political Economy, Sachs, J. D and Warner, A.M (997) Fundamental sources of long-run growth, The American Economic Review, 87, Sachs, J. D and Warner, A.M (200) The curse of natural resources, European Economic Review, 4-6, Appendix Proof of proposition : Consider an increase in S, nothing will change in equation (33) when ; and this is because nor prices of industrial goods neither expenditures are a ected by this change in the endowments. The case of the prices need not demonstration, this is because the primary good take no place in the production of industrial goods. We turn to see what happen with the expenditure in the region a ected by the boom. Making use of the de nition 23

24 of " and equations (7), (27), (46) and (48) we can express the total aggregate expenditure of region one as: [ + ( )] E L + w + p A Now, holding constant the wages, we derive this expression with respect A + p p A + p A 0 S S Turning now to equation (33) with this result, the wages must remain equal. Because the resource is non-tradable, their owners must consume it all, so the price must fall until they are willing to do it, this means that their incomes will not change. Proof of proposition 2: Note that expressions (33) and (34) are valid in the case of < ; then, using the same reasoning as in Proposition (), we can arrive to the same result for the wages (w ). But, what happen when S S 2 increases? Using the de nition of X we have (w w 2 (S S 2 ) ( ) w w 2 S S 2 > 0 The ratio of wages must increase. Why this happen? The boom has not e ect over the expenditures of the regions, as before. However, it will a ect the price-ratio of industrial goods, because the primary good is used for industrial production. Since the price of the resource reduces due to the increase in the supply, all rms in region can bene t from this lower primary price. To see this, lets take the derivative of the price-ratio holding constant the ratio of wages (for (p p 2 (S S 2 ) ( ) p p (p A p A2 ) < 0 p A p (S S 2 ) This reduction in the price-ratio will lead to an increase in the demand of the industrial goods of region and a decrease of the demand of goods from region 2. Again, we can see this (holding constant the ratio of wages) looking at the following derivatives: 24