Structural Change and Growth in a NEG model

Transcription

1 Structural Change and Growth in a NEG model Fabio Cerina CRENoS and University of Cagliari Francesco Mureddu CRENoS and University of Cagliari Preliminary draft, do not quote or circulate Abstract This paper presents a footloose capital model of structural change, agglomeration and growth. By assuming the same non-homothetic preference structure as Murata (2008, we obtain similar results in that a progressive reduction of trade costs allows the economy to pass from a pre-industrialized to an industrialized stage and then, within the latter, from a dispersed to an urbanized regime. However, the introduction of capital accumulation and the dynamic setting of our model opens the door to a richer set of implications. First, as far as the dynamics of equilibrium allocations are concerned, an additional stage is introduced as, for some intermediate values of trade costs, multiple equilibria regime emerges with the symmetric and the core-periphery equilibria stable at the same time. Second, as far as growth analysis is concerned, the introduction of non-homotheticity allows growth to depend on both trade costs and agglomeration even in the absence of localized spillovers. In particular, integration is always growth-enhancing while agglomeration is growth-detrimental. We believe our results to have important policy implications and to provide some natural mechanisms to explain a number of real-economies phenomena. Introduction A well known stylized fact in developed economies is the structural change, i.e. the phenomenon according to which the economic development has determined a shift in labour and expenditure allocation from agriculture to non-agricultural sectors and, within the latter, from manufactures to services. The recent literature highlights two main dierent strands of explanation for this stylized fact. One is supply based: the evolution of expenditure and labor shares are the result of dierences in sectoral TFP growth (Hansen and Prescott (2002; Ngai and Pissarides (2007 and Acemoglu and Guerreri (2008 among others. The other is demand based, so that the shift in the expenditure and labor shares is a result of non-homothetic preferences (Matsuyama (992; Laitner (2000; Caselli and Coleman (200; Gollin, Parente, and Rogerson (2002; Kongsamut, Rebelo and Xie (200 and Foellmi and Zweimuller (2008 among many others. Among the latter ones, Murata (2008 introduces non-homothetic preferences and decreasing returns to scale in the agricultural sector in a static model of New Economic Geography in order to show that integration can explain both demand-driven structural change and the urbanization process. Our model might be seen as a rst tentative to integrate the former approaches. First, it encompasses the spatial (agglomeration and trade costs and the dynamic (endogenous growth dimensions. Second, by introducing non-homothetic preferences and jointly allowing for productivity growth dierences within sectors, it encompasses the demand and the supply-based approaches. By introducing non-homothetic We would like to thank all the attendants to the seminar in Cagliari. Francesco Mureddu gratefully acknowledges nancial support from the European Commission and from Sardinia Autonomous Region (FSE , L.R.7/2007 "Promotion of scientic research and technological innovation in Sardinia". Corresponding author: Fabio Cerina, Viale Sant'Ignazio Cagliari (Italy - fcerina@unica.it. The concept of structural change has been pioneered by Fisher (939 and Clark (940, who state that the patterns of production are functions of the level of income and that resource and production shifts are part of development. Moreover according to Rostow (960 the economy passes through various stages of development from the traditional stage to the take-o stage to the mass consumption stage. Finally Kuznets (966, Chenery and Syrquin (975 argue that as the economy grows, production shifts from the primary (agriculture to the secondary (manufacturing to the tertiary sector (services.

2 preferences in a footloose capital model a lï¾ Baldwin and Martin (2004, we replicate the economic mechanism introduced by the static model of Murata (2008 and we analyse its implications in a dynamic framework of endogenous growth, which appears to be more proper when dealing with long-term phenomena like structural change. The structural change mechanism in our model is then led by the endogenization of sectoral expenditure shares. The latter ones will depend, in equilibrium, from both trade costs (as in Murata and the degree of spatial concentration of economic activities. Both components have an eect on real income through the price index of high-productivity good. Thanks to the non-unitary income elasticity due to non-homotheticity, such price changes will lead to a reallocation in the sectoral expenditure ( Engel's law and labor shares (Petty's law. But the two components (trade costs and degree of agglomeration act dierently in dierent regions. While a reduction in trade costs always increase both domestic and foreign purchasing power (leading to an expenditure and labor shift from the agricultural to industrial sector, not surprisingly spatial concentration of economic activities will have non-symmetric eects in the two regions. By home-market eect, domestic industrialization will - ceteris paribus - increase domestic real income and decrease foreign one. This mechanism is the source of the main results of our paper. These can be divided in two groups: the dynamic properties of equilibrium allocation; 2 the equilibrium growth prospect. As for the rst set of results, the non-unitary demand elasticity of goods with respect to income gives rise to a much richer and complex equilibrium dynamics with respect to the existing literature. First of all, non-homotheticity introduces a nonlinear element in the set of equilibrium conditions which lead to the emergence, for some intermediate values of trade costs, to multiple equilibria regime with the symmetric and the core-periphery equilibria stable at the same time. More precisely, among the many possible cases (described in detail in the rest of the paper there is a wide range of plausible parameter values such that the stability map with respect to (exogenous movements in the trade costs looks as follows. For almost prohibitive trade costs, prices can be so high that real income is low enough for the representative consumer to consume only the agricultural (and necessary good where no signicant technological progress is allowed. As a consequence, consistently with the so-called Malthusian era 2, growth is nil. Following Murata (2008 we call this a pre-industrialized or rural stage of the economy. Once trade costs becomes lower than a certain threshold, real income increases accordingly up to a level which allows the industrial (and luxury good to be consumed. This creates an incentive for industrial rms to set-up and invest in R&D. Hence the economy enters an early industrialized pre-urban stage with positive but low growth (trade costs are not low enough to allow for massive consumption of the technology-intensive good and with low degree of agglomeration (trade costs are not low enough to activate self-reinforcing mechanisms of agglomeration. When trade costs overcome a third threshold from below, two unstable interior non-symmetric steady-states emerge. As a consequence, both the dispersed (symmetric and the concentrated (core-periphery equilibria are stable at the same time. That means that a small perturbation for an economy located in one of these two equilibria may lead to self-reinforcing mechanism of both dispersion or agglomeration. As far as the equilibrium dynamics are concerned, this is a novel result with respect to Murata (2008. However, being the symmetric equilibrium still stable, an economy located in its neighborhood will fail to activate the urbanization process. But if for any (exogenous reason, the economy locates itself in one of the two unstable equilibria, then processes of catastrophic agglomeration or dispersion might be activated. We call this stage an intermediate industrial economy with either dispersion or agglomeration. Finally, when trade costs overcome a fourth threshold and become suciently close to zero, the symmetric equilibrium looses its stability and there is a strong incentive for industrial rms to concentrate in space to fully exploit increasing returns to scale and the proximity of a large market. We then nally enter a modern industrialized urban economy with a urbanized and industrialized core and a rural periphery. Unlike Murata (2008, where labor is mobile across regions and workers choose the location where real wage is higher, our self-reinforcing mechanism abstracts from workers' movements between regions and the shape of the economic geography is the result of industrial rms' localization decisions. An industrial rm chooses where to set-up a new unit of capital according to where prots are higher. When capital is immobile 3 (and can therefore be interpreted as human capital or skilled labor, these 2 Galor and Moav ( Our model basically assumes capital immobility, but we can easily derive the implications for the case of capital mobility, 2

3 prots are not repatriated but they are spent locally hence enlarging the size of the market and giving further incentives to industrial rms to locate in the same region. When trade costs are low enough, this typical demand-linked and self-reinforcing agglomeration force becomes stronger than the usual congestion eect which represents a dispersion force working on the opposite direction. But thanks to the endogenous expenditure shares, our model displays an additional demand-driven self-reinforcing mechanism of agglomeration: a reduction in trade costs, by decreasing the northern price index and increasing the northern real income, will lead to an increase in northern expenditure and labor shares in the industrial good, therefore amplifying the incentive for another rm to invest in the north. This additional agglomeration force, which is not present in standard models of New Economic Geography and Growth, has the same nature of the one introduced by Cerina and Mureddu (20, and we call it Expenditure shares eect 4. The second set of results are related to the long-run equilibrium growth patterns. As the growth rate is positively aected by the expenditure shares in the faster productive sector, non-homothetic preferences allows growth to depend - in contrast with the standard models - on both trade costs and agglomeration even in the absence of localized spillovers. As for the role of integration, it is easy to show that integration, by always increasing both regional expenditure shares on industrial goods, always creates an additional incentive for industrial rms to invest in R&D and, therefore, is eventually good for growth. In doing so, our model provides an alternative channel (briey introduced in Cerina and Mureddu (20 through which integration boosts growth 5 with respect to the traditional explanations based on comparative advantage, technology ows and eciency gains. As for the role of agglomeration, it turns out to be always growth-detrimental because it reduces the average aggregate expenditure share in industrial goods. This provides an explanation (alternative to Cerina and Mureddu (2009 for the recent empirical evidence according to which agglomeration boosts GDP growth only up to a certain level of economic development (Bruhlardt and Sbergami (2009. The opposite eect that integration and agglomeration have on growth, together with the fact that agglomeration itself is aected by integration, gives us a good reason to associate a growth analysis to each stages of equilibrium industrial allocation described earlier. Our main nding in these regard is that the positive eect of integration on growth will be higher (in absolute terms than the negative eect of agglomeration. In other words, despite the fact that more integration leads to more agglomeration which in itself reduces growth, an hypothetical central planner willing to maximize growth at the aggregate level will always choose the maximum level of integration. Given the appeal that NEG model has on policy-makers, we believe our results to have important policy implications as, for some plausible range of parameters' values, they are opposite to the commonly accepted models. The rest of the paper is organized as follow. Section 2 presents the analytical framework; section 3 deals with the dynamic properties of equilibrium allocations; section 4 is dedicated to the analysis of the rate of growth and section 5 concludes. 2 The analytical framework The analytical framework of the paper can be considered as a mix of Baldwin, Martin and Ottaviano (200 and Murata (2008. which is simply a special case of the former. 4 However, unlike the latter and other papers, this force is shown to be so strong to allow for the possibility of catastrophic agglomeration even when trade costs are prohibitive and even in the case of capital mobility. 5 The positive relationship between integration and growth is widely accepted. Edwards (998 showed that, out of nine indicators of trade openness (taris, non-tari barriers, etc., eight were positively related to TFP growth in a sample of 93 countries. Dollar (992 argued that an indicator of openness based on price deviations was positively associated with growth. Sachs and Warner (995 showed that less open countries experienced annual growth rates 2 % below open ones in Frankel and Romer (999 estimated that a % point increase in the trade to GDP ratio causes almost a 2 % increase in the level of per capita income, and nally Wacziarg and Welch (2008 show that episodes of trade liberalization are followed by an average increase in growth on the order of -.5 % points per annum. 3

4 2. Production side There are two regions, North and South, which are symmetric in terms of production factors (labour L and capital K, preferences, technology, trade costs and labour endowments. In both regions three sectors are active: traditional T, manufacturing M and innovation I. Labour is immobile across regions but mobile across sectors. the traditional good T is produced under perfect competition and constant returns to scale so that it's production function is T = L. Moreover it is freely traded, while the manufacture good M is subject to iceberg trade costs (Samuelson (954. Manufactures are produced under Dixit-Stiglitz monopolistic competition (Dixit and Stiglitz (977 and enjoy increasing returns to scale: rms face a xed cost in terms of knowledge capital 6 and a variable cost a M in terms of labour. Thereby the cost function is π + wa M x i, where π is the rental rate of capital, w is the wage rate and a M are the unit of labour necessary to produce a unit of output x i. Since a unit of capital K is required in order to set up the production of a new variety n, we have K + K = K w = n + n = n w. When capital is immobile, once a rm is set-up in a given region, the owner of the capital unit is forced to spend its capital income in the same region, therefore increasing the local market-size. This is not the case when capital is mobile, so that a production shift is not associated to a demand shift. In this paper we will basically assume capital immobility but the case of capital mobility, being a special case of the former, will be briey analysed along the text. In case of capital immobility (alike Baldwin, Martin and Ottaviano (200: n = K and n = K. By dening s n = and s K = K K, we also have s w n = s K. Each region's K is produced by its I-sector, so the production and the cost function of innovation are respectively: K = L I, F = w I a I a I To individual I-rms, the innovation cost a I is a parameter, thereby our model enjoys endogenous growth assuming that the labour unit requirements a I decline as the cumulative output rises. In this model learning spillovers are assumed to be global 7 so that: a I = K w The growth rate of capital, rms and varieties is then given by: g K K ; g In the following analysis, we will focus on the northern region, as the southern expressions and denitions are isomorphic. 2.2 Households' behaviour As for the demand-side, an innitely-lived representative household maximizes: U = ˆ t=0 K K e ρt ln Qdt; Q = (C A µ ( n w C M + γ µ ; C M = (ˆ K+K i=0 c i di Where ρ > 0 is the time-preference rate, γ > 0 is the non-homotheticity parameter, Q is the consumption bundle which is a Stone-Geary non-homothetic mixture of the manufacture bundle C M and of the agricultural good C A. Finally, the manufacture bundle C M is a mixture of the n w industrial varieties. The preference structure is basically the same as in Murata (2008 except for the intertemporal dimension which is not present in the latter. Alternatively, it can be viewed as the same preference structure of Baldwin, Martin and Ottaviano (200 once γ is set to zero. The latter introduces a nonhomothetic element in the utility function which makes the demand elasticity with respect to (nominal 6 It is assumed that producing a variety requires a unit of knowledge interpreted as a blueprint, an idea, a new technology, a patent, or a machinery. When capital is immobile, then a unit of K might also be interpreted as a mix of physical and human capital (Baldwin and Martin ( The analysis can be also developed with localized knowledge spillovers, but it will divert the attention from the mechanism we aim to highlight n n w 4

5 expenditure to be larger than for industrial goods and smaller than for agricultural goods. This deviation from the standard model is the source of all the results of this paper. Also notice that, as in Murata (2008 in the context of a NEG model and Blanchard and Kiyotaki (987 in a macroeconomic context, we neutralize agents' love for variety by setting to zero its parameter. An analytical consequence of abstracting from the love of variety is the emergence of the term n w in the second-stage utility function: this normalization neutralizes the dependence of the price index on the number of varieties allowing us to concentrate the analysis on the inuence of rms' location and transport costs on the expenditure shares. By abstracting from the love of variety, we are able to focus on the eect that a non-unitary value of the demand elasticity with respect to income has on the equilibrium outcomes of the model by still preserving constant expenditure shares in equilibrium. The innitely-lived representative consumer's optimization is carried out in three stages. In the rst stage the agent intertemporally allocates consumption between expenditure and savings. In the second stage she allocates expenditure between manufacture and traditional goods, while in the last stage she allocates manufacture expenditure across varieties. As a result of the intertemporal optimization program, the path of consumption expenditure E across time is given by the standard Euler equation: Ė E = r ρ ( with the interest rate r satisfying the no-arbitrage-opportunity condition between investment in the safe asset and capital accumulation: r = π F + F F where π is the rental rate of capital and F its asset value which, due to perfect competition in the I-sector, is equal to its marginal cost of production. In the second stage the agent chooses how to allocate the expenditure between manufacture and the traditional good according to the following optimization program: µ max Q t = (C A (n µ w C M + γ C M,C T s.t. : P M C M + p A C A = E where p A is the price of the traditional good (which is chosen as numeraire good and P M = [ K+K p i=0 i ] di is the Dixit-Stiglitz perfect price index. As a result of the maximization, and by setting p A = we obtain the following demand for the manufactured and the traditional goods: P M C M E = max C A E = min ( ( µ ( µ γ µ + ( µ γ P M En w P M ; 0 En w ; = max m nw = min m ; 0 E/P M nw ; E/P M These expressions deserve some comments. First, notice that once γ is set to zero, the expenditure shares turn to collapse to the usual Cobb-Douglas ( one. Second, it is easy to see that when γ is large enough and/or E/P M are low enough, then m n w E/P M can be negative so that consumption of industrial goods is set to zero and, consequently, expenditure shares in the agricultural good are equal to one. We then have: m P M < 0 m E > 0 5

6 so that regional expenditure share in manufacture rises with regional total real expenditure. In other words, industrial good is a luxury good while( agricultural good is a necessary good. Finally, in the third stage, whenever m n w E/P M is positive, the amount of M goods expenditure is allocated ( across varieties according to the a CES demand function for a typical M variety c j = p j P M m are isomorphic. n w E/P M E, where p j is variety j's consumer price. Southern optimization conditions 2.3 Expenditure shares, integration and agglomeration Due to perfect competition in the A-sector, the price of the agricultural good must be equal to the wage of the traditional sector's workers: p A = w A. Moreover, as long as both regions produce some T, the assumption of free trade in T implies that not only price, but also wages are equalized across regions. It is therefore convenient to choose home labour as numeraire so that: p A = p A = w A = w A = As it is well-known, it's not always the case that both regions produce some T. An assumption is actually needed in order to avoid complete specialization: a single country's labour endowment must be insucient to meet global demand. Formally, we need: L < m nw E + m nw E/P M E /PM E The purpose of making this assumption, which is standard in most New Economic Geography and Growth models 8, is to maintain the M sector and the I-sector wages xed at the unit value: since labour is mobile across sector, as long as the T - sector is present in both regions, a simple arbitrage condition suggests that wages of the three sectors cannot dier. Hence, M sector and I-sector wages are tied to T -sector wages which, in turn, remain xed at the level of the unit price of a traditional good. Therefore: w M = w M = w T = w T = w = (2 Finally, since wages are uniform and all varieties' demands have the same constant elasticity, rms' prot maximization yields local and export prices that are identical for all varieties no matter where they are produced: p = wa M. Then, imposing the standard normalization which assigns the value to the marginal labor unit requirement and using (2, we nally obtain: p = w = (3 As usual, since trade in the M good is impeded by iceberg import barriers, prices for markets abroad are higher: p = τp; τ By labeling as p ij M the price of a particular variety produced in region i and sold in region j (so that p ij = τp ii and by imposing p =, the M goods price indexes might be expressed as follows: P M = P M = [ˆ n 0 [ˆ n 0 (p NN M di + (p NS M di + ˆ n 0 ˆ n 0 (p SN M di (p SS M di ] ] = (s K + ( s K φ n w (4 = (φs K + s K n w (5 8 See Bellone and Maupertuis (2003 and Andrï¾ s (2007 for an analysis of the implications of removing this assumption. 6

7 where φ = τ is the so called "phi-ness of trade" which ranges from 0 (prohibitive trade to (costless trade. Substituting the new expressions for the M goods price indexes in the northern and southern M goods expenditure shares, and considering that E = s E E w and E = ( s E E w where E w is world expenditure and s E is northern share or world expenditure (or northern market size, regional expenditure shares on industrial goods can be written as: P M C M E P M C M E = max = max ( ( µ ( µ γs E (s K + ( s K φ E w ; 0 µ ( µ γ ( s E (φs K + s K E w ; 0 = max (0, m (s K, φ, s E, E w (6 = max (0, m (s K, φ, s E, E w (7 Since φ, γ and µ are parameters and, by denition, s E and s K should be constant in equilibrium, then the only source of variation for regional industrial goods expenditure shares in equilibrium can only be global world expenditure E w. We now show this is not the case because E w is constant too in equilibrium. We start considering that labour force at world level is given by the sum of the labour forces employed in the three sectors (innovation, manufacture and traditional in both regions: 2L = L I + L I + L M + L M + L T + L T Assume industrial expenditure shares are strictly positive so that an industrial sector exists. In this case we know that the world sectoral consumers' expenditure should be equal to the sectoral value of total production, so that: L M + L M = (m (s K, φ, s E, E w s E + m (s K, φ, s E, E w ( s E E w (8 L T + L T = [( m (s K, φ, s E, E w s E + ( m (s K, φ, s E, E w ( s E ] E w (9 Combining those three equations we nd a relationship between E w, s E, s K and φ: E w = (2L L I L I s E ( m (s K, φ, s E, E w + ( s E ( m (s K, φ, s E, E w (0 Notice now that, given the knowledge capital production function stemming from the innovation sector, we can write: L I = Ka i = K K w = K K K w K = gs K L I = K a i = K K w = K K K w K = g ( s K Hence, in equilibrium, g = g = L I + L I, so that: E w = (2L g s E ( m (s K, φ, s E, E w + ( s E ( m (s K, φ, s E, E w ( solving this xed point problem, we nd a solution for E w as function of parameters (L,, µ, φ, γ and µ and variables which are constant in equilibrium (s E, s K, g: E w = (2L g ( µ ] ( µ γ [( s E 2 (s K + ( s K φ + s 2 E (φs K + s K ( µ (s K + ( s K φ (φs K + s K (2 Hence world expenditure is constant in equilibrium. Since in equilibrium s E is constant too by denition, even regional expenditures E and E must be constant. That means that, from the Euler 7

8 equation (, r = ρ so that the interest rate should be equal to the discount rate. An implication for this result is that: r = π F + F F = ρ but since F F = g and F = a I = K w we also have: ρ + g = πk w This expression helps us in nding a formulation for E and E which takes into account the so-called market-size condition. By national account, regional expenditure is equal to the sum of regional factor income (wage bill wl = L and operating prots πk and π K minus regional spending on new capital L I and L I. Hence we have: E = L + πk L I = L + πk w s K gs K E = L + π K L I = L + π K w ( s K g ( s K Now, if s K is an interior equilibrium, then it should be π = π. By contrast, if s K = (or 0 then πs K = π (or π ( s K = π. In any case, there is only one operating prot so that π = π. Hence we have π K w = πk w = ρ + g. Finally, by substituting and collecting terms we nd: E = L + ρs K E = L + ρ ( s K As a consequence, we also have a simple expression for world expenditure and for the market-size condition: E w = 2L + ρ s E = L + ρs K 2L + ρ Substituting for this two values in the expressions for m (s K, φ, s E, E w and m (s K, φ, s E, E w we nally obtain the expressions for regional expenditure shares in industrial goods as functions of trade costs φ and industrial allocation s K : P M C M E P M C M E = max (µ ( µ γ (s K + ( s K φ ; 0 = max [m (s K, φ, 0] (3 L + ρs K = max (µ ( µ γ (φs K + ( s K ; 0 = max [m (s K, φ, 0] (4 L + ρ ( s K We now focus on these set of expressions, which represents the core of the main results of our paper. We rst analyse the role of industrial allocation s K in determining the equilibrium regional industrial expenditure shares. Unlike models with homothetic second stage-utility, northern and southern expenditure shares in manufactures can dier when s K /2. In particular, it is easy to see that: m = ( µ γ (s K + ( s K φ ( φ (L + ρs K + (s K + ( s K φ ρ (L + ρs K 2 > 0 m = ( µ γ ( φ (φs K + ( s K (L + ρ ( s K + ρ (φs K + ( s K (L + ρ ( s K 2 < 0 So that an increase in the share of industrial rms located in the North increases northern expenditure shares and reduces southern expenditure shares. This is due to a twofold eect. 8

9 The rst eect works through the regional price indexes P M and PM : by home market eect, an increase in s K allows northern households to purchase a lower amount of goods from the other region so that they are less hurt by trade costs. This reduces northern industrial price index and then increases northern real income which we know has a positive eects in industrial expenditure shares (see equation (6. The opposite, of course, happens in the South. The second eect works through regional global expenditures E and E : an increase in s K to an increase of northern operating prots which is a determinant of northern global expenditure. But since the elasticity of demand for industrial goods with respect to expenditure is higher than, this leads to an increase in northern industrial expenditure shares and a correspondent reduction in southern industrial expenditure share. These two eects also explain in which sense agglomeration leads to Engel's Law. As for the role of trade costs, their eects are more symmetric in the two regions: a higher degree of integration (larger φ leads to lower prices in both regions and then, ceteris paribus, to an increase in North and South purchasing power. This increase - whose exact amount will dier in the two regions as long as s K /2 - allows the representative household to allocate a larger fraction of expenditure in the luxury good, manufactures. Formally we have: m φ m φ = = ( µ γ (s K + ( s K φ ( s K > 0 L + ρs K ( µ γ L + ρ ( s K (φs K + ( s K s K > 0 Hence, as in Murata (2008, a reduction in trade costs is also able to explain the observed Engel's law. Finally, it is important to highlight that it is not always the case that expenditure shares are strictly positive: there is a wide range of values of trade costs φ, industrial allocation s K and other parameters for which the representative household does not nd it optimal to consume the industrial goods. For this range of parameters' values, the industrial sector simply does not exist and the economy is in a pre-industrial stage. Section 3 will analyse this phenomenon Petty's law As in Murata (2008, agglomeration and integration in our model are also able to explain the so-called Petty's law, that is the observed shift of the labour force from agricultural to non-agricultural sectors. By using m(s K, φ and m (s K, φ in (8 and (9 we can write: and: L T + L T = ( m (s K, φ (L + ρs K + ( m (s K, φ (L + ρ ( s K L M + L M = (m (s K, φ (L + ρs K + m (s K, φ (L + ρ ( s K By dierentiating the latter with respect to φ we nd: (L M + L M = ( m (sk, φ (L + ρs K + m (s K, φ (L + ρ ( s K φ φ φ which is clearly positive. Hence, considering that the both northern and southern labor forces are constant at, respectively, L and L, we conclude that the integration explains Petty's law both at the regional and aggregate level. The same cannot be said about agglomeration. As for the northern industrial labour force we nd: while: L M = ( m (sk, φ (L + ρs K + ρm > 0 L M = ( m (s K, φ (L + ρ ( s K ρm < 0 9

10 So that, not surprisingly, agglomeration in the North has an opposite eect on northern industrial labor force (positive and southern industrial labor force (negative. Of course, the net aggregate eect of agglomeration on the global industrial labour force will depend on the relative intensity of the two regional eects. 3 Integration, industrialization and agglomeration In the following section, we will perform a detailed analysis of the equilibrium dynamics of industrial allocation. As the only source of the dynamics in our model is the birth of new industrial rms, the dynamic analysis can only be undertaken if the industrial sector eectively exists. However we have seen that this is not always the case as with non-homothetic preferences there is a wide range of parameters values such that the consumer demand for industrial goods is nil and therefore there is no incentive for industrial rms to set a new investment. The rst part of this section is then dedicated to the analysis of the conditions under which what we call a pre-industrial economy - i.e. an economy where the whole consumer demand is devoted to agricultural goods - is a steady state. Then we address a question about the number of feasible interior steady-states that our economy is able to exhibit and how the number of admissible steady-states change with the model's parameters. Finally, we will study the dynamic properties of any steady state and we will perform a stability map in order to see how the dynamic properties of the economy change as trade costs decreases. The nal contribution of this section will be that an increase in the degree of integration (as suggested by the data might be responsible, under reasonable parameters' restrictions, of the evolution of the economy through 4 dierent stages: a pre-industrial stage with no demand for manufacturing goods, a symmetric early-industrial stage without agglomeration, a multiple equilibria regime with or without agglomeration and, nally, a modern-industrialized economy with agglomeration. 3. A pre-industrialized economy As we can easily see from (3 and (4 it is not always the case that regional expenditure shares in manufactures are strictly positive. In particular: P M C M E = 0 µ ( µ γ (s K + ( s K φ L + ρs K P M C M E = 0 µ ( µ γ (φs K + ( s K L + ρ ( s K In this case, given the lack for demand for manufactures, the industrial sector simply does not exist and the regional workforce is wholly allocated to the agricultural sector. That could happen when: the importance of industrial goods in the utility function µ is low enough; 2 the non-homotheticity parameter γ is large enough; 3 the number of workers L and the interest rate ρ (which positively aect nominal expenditure are small enough and, most importantly, 4 trade costs (which negatively aect purchasing power are small enough. Therefore, industrialization can be triggered by an exogenous change of one of the previous parameters. Following the NEG tradition, we concentrate on the eect of an exogenous reduction in trade costs, which will be more deeply analysed later. When trade costs are suciently high - and so φ is close enough to zero - there might be no demand for industrial goods and therefore no incentive for an industrial rm to set-up. In this case, the economy is in a pre-industrial stage of development. In such a stage, the level of trade costs should be such that: φ < φ I : m (s K, φ = max (µ ( µ γ (s K + ( s K φ ; 0 = 0 L + ρs K φ < φ I : m (s K, φ = max (µ ( µ γ (φs K + ( s K ; 0 = 0 L + ρ ( s K 0

11 where φ I and φ I are the level of freeness of trade below which industrialization is not triggered in the North and in the South respectively. Given the perfect symmetry of the two regions in terms of technology, endowments and preferences, industrialization should be triggered simultaneously in the two regions because, in a rural economy, the incentive to set-up a new rm cannot dier across regions. There are two consequences for this observation: rst φ I = φ I because trade-costs are symmetric; second, at the very beginning of the industrial age (i.e. when φ = φ I the industrial sector should be equally divided among the two regions so that the rstindustrial steady state (stable or not is a symmetric one. As a consequence, φ = φ I implies s K = 2. Therefore we have: ( ( ( +φi 2 m 2, φi = m, φi 2 = µ 2 ( µ γ 2L + ρ From this expression we can nd an explicit value for φ I, i.e., the industrialization-triggering level of freeness of trade which is: ( 2 ( µ γ φ I = 2 µ (2L + ρ Notice that there are ranges of parameters' values such that φ I is either negative or larger than. In the rst case, the economy is always industrialized for whatever values of trade costs. That happens when: ( µ γ < = 0 ( ( µ L + ρ ( 2 ( µ γ φ I = 2 < µ (2L + ρ In the second case, industrialization is ruled out for any values of trade costs. That happens when: ( ( µ γ > µ L + ρ ( 2 ( µ γ φ I = 2 > 2 µ (2L + ρ Clearly, if industrialization is impossible, there is no room for agglomeration. On the other hand, when φ I < 0, industrialization trivially exists for any level of trade costs. For this reason, in the rest of the paper we concentrate on the case when φ I belongs to the interval (0,. A rural economy is then a steady state when φ < φ I. What happens when φ > φ I? This will be the topic of the next subsections. We will rst investigate the existence, the nature and the number of steady states in an industrial economy, then we will analyse their stability properties and, nally, we will provide a stability map to see how the spatial patterns of our economy changes as trade costs exogenously decrease. 3.2 An industrial economy: Tobin's q and Steady-State Allocations In the next subsections, we will assume that φ > φ I so that both northern and southern households' demand for industrial goods is strictly positive and industrial rms have incentive to set-up new investments. What does a steady state look like in such an economy? In any interior steady state the growth rate of the world capital stock will be constant and will either be common (g = g in the interior case or north's (g > g = 0 in the core-periphery case 9. In any case, the value of investing in a new unit of capital in steady state is respectively in the two regions: 9 Time-dierentiating s K = K K W V = π ρ + g ; V = π ρ + g we obtain the dynamics of the share of manufacturing allocated in the north: ( K ṡ K = s K ( s K K K K As we can see only two kinds of steady states (ṡ K = 0 are possible: a steady-state in which the rate of growth of capital is equalized across countries (g = g ; 2 a steady state in which the manufacturing industries are allocated and grow in only one region (s K = 0 or s K =

12 The expressions for prots stemming from rms' prot maximization and iceberg trade costs are: Where: B (s E, s K, φ = B (s E, s K, φ = [ [ π = B (s E, s K, φ Ew K w π = B (s E, s K, φ Ew K w s E (s K + ( s K φ m (s K, φ + φ ( s E s E φ (s K + ( s K φ m (s K, φ + ] (φs K + s K m (s K, φ ] s E (φs K + s K m (s K, φ By using the labour market clearing condition and the expressions for the prots we are able to nd the equations representing the Tobin's q in the two regions: q = V F = B (s E, s K, φ E w (ρ + g q = V F = E w B (s E, s K, φ (ρ + g Each rm will invest in the more protable regions, i.e. where the Tobin's q is higher. Since rms can be created both in the north and in the south, the following condition must hold in any interior equilibria: q = q = The rst equality is a no-arbitrage condition (q = q, stating that in any interior equilibrium there will be no incentive for a rm to move to another region. The fact that both regions' q should be equal to, represents the optimal investment condition, according to which in equilibrium rms will decide to invest up to the level at which the expected discounted value of the rm itself is equal to the replacement cost of capital. By solving this equation we nd the steady-state relation between the northern market size s E and the northern share of rms s K : s N E (s K, φ = m (s K, φ (s K + ( s K φ m (s K, φ (φs K + s K + m (s K, φ (s K + ( s K φ The denition of s E when labour markets clear gives us the permanent income condition, which is a relation between northern market size s E and the share of rms owned by northern entrepreneurs s K : s P E (s K = L + ρs K 2L + ρ Those two relations drive the dynamics of our economy. Thereby we can dene a new implicit function whose zeros represent the interior steady state allocations of our economy: f (s K, φ = s N E (s K, φ s P E (s K We dene an interior steady state allocation as any value of s K (0, such that f (s K, φ = 0. Notice that the symmetric allocation ( ( s K = 2 is always an equilibrium, because f 2, φ = 2 2 = 0. However, and unlike most NEG models, this is not the only feasible interior steady state as the non-linearity of s N E (s K, φ opens the door to multiple intersections with the linear function s P E (s K and therefore to dierent value of s K such that f (s K, φ is zero. We now explore this issue in detail. 2

13 3.3 Interior Steady States In this section we provide the necessary and sucient conditions according to which the steady state turns to be unique or threefold. Consider f (s K, φ, which can also be written as the following: f (s K, φ = m (s K, φ (s K + ( s K φ m (s K, φ (φs K + s K + m (s K, φ (s K + ( s K φ L + ρs K 2L + ρ where m (s K, φ and m (s K, φ are respectively the expenditure shares in manufacture in the north and in the south: m (s K, φ = µ ( µ γ (s K + ( s K φ L + ρs K m (s K, φ = µ ( µ γ (φs K + ( s K L + ρ ( s K These expenditure shares are positive as long as, by assumption, φ > φ I. By substituting for the explicit form of the expenditure share, we nd: f (s K, φ = µ (s K + ( s K φ [L + ρs K ] [L + ρ ( s K ] ( µ γ (φs K + ( s K (s K + ( s K φ [L + ρs K ] µ ( + φ [L + ρs K ] [L + ρ ( s K ] ( µ γk (s K, φ where: k (s K, φ = (s K + ( s K φ (φs K + ( s K ] [(s K + ( s K φ [L + ρ ( s K ] + [L + ρs K ] (φs K + ( s K Notice that is f (s K, φ symmetric with respect to the point ( 2, f ( 2 meaning that f (sk = f ( s K. This symmetry is very important as it allows us to limit the analysis to the interval s K [ [ 0; 2 and then extend it to the rest of the feasible values of s K 2, ], by simply applying the symmetry rule. Dene now the function: h (s K, φ = f (s K, φ N (s K, φ = µ ( 2s K (ρφ L ( φ ( µ γ where: N (s K, φ = [ (φs K + ( s K (s K + ( s K φ (s K + ( s K φ (φs K + s K (5 (L + ρs K (L + ρ ( s K (2L + ρ [µ ( + φ [L + ρs K ] [L + ρ ( s K ] ( µ γk (s K, φ] > 0 Since N (s K, φ, s K [0, ] and for any φ > φ I, we have that f (s K, φ = 0 h (s K, φ = 0: every zero of h (s K, φ is also an interior steady state and vice-versa. In particular, it is easy to see that h ( 2, φ = 0. We can then re-write f (s K, φ as: By dierentiating with respect to s K we nd: f (s K, φ = h (s K, φ N (s K, φ f (s K, φ = h (s K, φ N (s K, φ + h (s K, φ N (s K, φ ] but, as we have seen: ( h 2, φ = 0 3

14 so that: f ( 2, φ = h ( 2, φ ( N 2, φ thereby we can conclude that: [ ( f 2 sign, φ ] [ ( h 2 = sign, φ ] so that, in the symmetric allocation, the sign of the slope of f is the same as the sign of the slope of h. This is also an important property which allows us to concentrate on h (s K, φ which is much easier to deal with from the mathematical point of view and it is of great help in the proof of the following proposition. Proposition (Number of interior steady states The system displays one or three interior steady state allocations: the symmetric allocation s K = 2 (which is a "global" interior steady state and two nonsymmetric allocations: s K (L, ρ, φ, γ and s K (L, ρ, φ, γ = s K (L, ρ, φ, γ which may emerge only for some values of the parameters. The symmetric steady state is unique when h (0, φ h(,φ 2 > 0 while there are 3 interior steady states when h (0, φ h(,φ 2 > 0. Proof. Please refer to the appendix This proposition provides a necessary and sucient condition for the uniqueness/multiplicity of interior steady states. It states that, given the monotonicity of h(,φ 2 in the interval [ 0, 2 and of the symmetry of h, uniqueness is guaranteed when h (0, φ, (i.e. the intercept of f in s K = 0 and h(,φ 2 (i.e. the slope of h( with respect to s K in the symmetric equilibrium have opposite sign. Despite its importance, proposition is not particularly informative as long as we do not provide an analysis concerning the way h (0, φ h(,φ 2 changes sign as trade costs decline. Because of the crucial linkages with the stability issues, such analysis will be performed in the section 3.5 together with the stability map. 3.4 Core-periphery steady states Interior steady-states are not the only allocation where the regional share of industrial rms is constant: s K is constant even when the latter is equal to either or 0, i-e., when the whole industrial sector is located in only one region. Since the two core-periphery allocations are perfectly symmetric, we just focus on the rst where the North gets the core. By following Baldwin and Martin (2004, we consider that for s K = to be an equilibrium, it must be that q = V/F = and q = V /F < for this distribution of capital ownership: continuous accumulation is protable in the north since V = F, but V < F so no southern agent would choose to setup a new rm. Dening the core-periphery equilibrium this way, it implies that it is stable whenever it exists. 3.5 Stability map: a new agglomeration force In this section we provide a complete stability map for the equilibria of our economy 0. As we will see, this analysis is intimately linked to the issue of the number of interior steady states. At the end of this section we will be able to state, for any value of the trade costs, the existence and stability of any kind of steady state (symmetric, non-symmetric or core-periphery. Following Baldwin and Martin ( This analysis is highly based on Cerina and Mureddu (20 section 3.4 where expenditure shares are endogenous as well but for a dierent reason: a non-unitary intersectoral elasticity of substitution between the traditional and the modern good. A more formal stability analysis, involving the study of the sign of the Jacobian associated to the dynamic system in E, E and s K, has been carried out and its results are identical to those reported in this section. Such calculations are available at request. 4

15 we consider the ratio of northern and southern Tobin's q: [ ] q q = B(s s E E, s K, φ B (s E, s K, φ = s K +( s K φ m(s K, φ + φ( s E φs K +( s K m (s K, φ [ ] = γ (s E, s K, φ (6 s E φ s K +( s K φ m(s s K, φ + E φs K +( s K m (s K, φ Starting from any interior steady-state allocation where γ (s E, s K, φ =, any increase (decrease in γ (s E, s K, φ will make investments in the North (South more protable and thus will lead to a production shifting to the North (South. Hence any allocation will be stable if a production shifting, say, to the north ( > 0 will reduce γ (s E, s K, φ. By contrast, if γ (s E, s K, φ will augment following an increase in s K, then the equilibrium will be unstable and agglomeration or dispersion processes might be activated. Taking the derivative of γ (s E, s K, φ with respect to s K and then using the no-arbitrage condition (which must be true in every interior steady state we nd: where: C (s K, φ = γ (s E (s K, s K, φ = A (s K, φ + B (s K, φ + C (s K, φ (7 A (s K, φ = ( m /m m ( φ /m ( + φ ( φ 2 B (s K, φ = (s K + ( s K φ (φs K + ( s K ( φ ds E (s K (m (φs K + ( s K + m (s K + ( s K φ 2 ( + φ ds K mm (s K + ( s K φ (φs K + ( s K : expenditure share eect : market crowding eect : demand eect The last two forces are the same we encounter in the standard New Economic Geography and Growth (NEGG models and they are the formal representation of, respectively, the market-crowding eect and the demand-linked eect. The rst force represents the novelty of our model. In the standard case, where m (s K, φ = m (s K, φ = m and then m = m = 0, this force simply does not exist. We dub this force as the expenditure share eect in order to highlight the link between the existence of this force and the fact that the expenditure shares are endogenous (thanks to a non-unitary value of the demand elasticities with respect to expenditure income. This expenditure share eect always represents a destabilizing force because 0 and m 0. Therefore, a new agglomeration force emerges thanks to the luxury nature of industrial (and more technologically-intensive good. But what is the economic intuition behind this new agglomeration force? Imagine a rm moving from south to north ( 0. For a given value of φ, this production shifting reduces the industrial good price index in the North and since demand elasticities with respect to income are non-unitary, this changes in the relative price, by also aecting the relative purchasing power, will also aect the regional values of the expenditure shares in the two kinds of good, creating an asymmetric structural change in the two regions. In particular, since manufacture is a luxury good, the increase in the Northern ( m purchasing power will also increase northern expenditure shares in the industrial good 0. The opposite will happen in the south since m 0. As we will see below, this new force can be so strong that a core-periphery outcome may be reached even in case of capital mobility and for whatever level of transport costs. More formally, any interior equilibria is stable (unstable when: By (7 and (?? that happens when: γ (s E, s K, φ (> 0. ( ( ds P E (s µµ K ρ φ 2 dµ ds K µ dµ ds K µ (s K + ( s K φ (φs K + ( s K = (> ds K 2L + ρ (µ (φs K + ( s K + µ (s K + ( s K φ 2 5 m

16 By computation we nd that: where: f (s K, φ = ( q q = ( φ [m (φs K + s K + m (s K + ( s K φ] 2 ( + φ (mm (φs K + s K (s K + ( s K φ [ f (s ] K, φ ( m m m m (φs K + s K (s K + ( s K φ + mm ( φ ( + φ [m (φs K + s K + m (s K + ( s K φ] 2 dsp E (s K ds K This proves the following proposition Proposition 2 In any interior equilibrium we have sign γ(s E,s K,φ interior steady state s K allocation is stable (unstable whenever: = sign f(s K,φ. Therefore each f(s K, φ (< 0 In other words, any interior equilibria is stable (unstable if the graph of f in the plane (s K, f (s K, φ crosses the horizontal axis with positive (negative inclination. Proposition 2 has several very important implications. The rst implication concerns the fact that the particular shape of the function f (and then h allows us to focus only on the value of this derivative in s K = 2 in order to deduce the stability properties of each (interior or core-periphery steady state. It is in fact straightforward to see, by proposition and by continuity and symmetry of f (and then h, that the sign of f(s K,φ in the symmetric equilibrium must be opposite to the sign of the same derivative in the two interior non-symmetric equilibria. More formally, if s K ( 0, 2 is a non-symmetric steady state for some φ, then we have: ( ( f(s K, φ f( 2, φ ( ( f( s = K, φ f( 2, φ < 0. (8 As a consequence, by proposition 2, the non-symmetric equilibria (when they exist are unstable when the symmetric equilibrium is stable and vice versa. By applying a similar reasoning we can conclude that s K = 0 and s K = are (local attractors, and therefore the two core-periphery equilibria exist, only when the non-symmetric interior steady states exist and are unstable or when the symmetric steady state is unique and unstable. The second implication is that the sign of f( 2,φ is not only informative on the stability of any kind of equilibria, but it is also a determinant of the uniqueness or multiplicity regime. It is therefore necessary to study how the sign of this derivative changes with the trade costs in order to gain simultaneous informations on the number of equilibria and on their stability as trade costs decline A special case: capital mobility Even if our model assumes capital immobility, we can briey analyse the case with capital mobility which is simply a particular case of the latter. When capital is mobile, prots are repatriated back home so there is no connection between the set up of a new rm (ds K and the variation in regional expenditure (ds E. In other words ds E ds K ( m m m m = 0 so that, with capital mobility, any allocation is stable if the quantity: (φs K + s K (s K + ( s K φ + mm ( φ ( + φ is negative. Notice that, when γ = 0, and then m m = m m, this can never be true. And in fact, with homothetic preferences every initial allocation is stable when capital is mobile. However, with non-homothetic preferences, any steady-state (in particular the symmetric one allocation might become unstable for suciently low trade costs even when capital is mobile. That can happen because when the expenditure eect is strong enough, a new industrial investment in the North might be able to generate an 6