Monopolistic competition, trade, and endogenous spatial fluctuations

Regional Science and Urban Economics 31 (2001) 51–77 www.elsevier.nl / locate / econbase

Monopolistic competition, trade, and endogenous spatial fluctuations Gianmarco I.P. Ottaviano* Universita` Commerciale ‘‘ L. Bocconi’’, CORE and CEPR, Istituto di Economia Politica, Via Gobbi 5, 20136 Milan, Italy Received 31 July 2000; received in revised form 25 September 2000; accepted 4 October 2000

Abstract This paper presents a model that exhibits the same richness of results as Krugman’s (1991b, Increasing returns and economic geography, Journal of Political Economy 99, 483–499) core–periphery model and is nonetheless reducible to a system of explicit differential equations. This makes it possible to provide an analytical investigation of the conditions that support the existence of multiple equilibria and endogenous spatial fluctuations driven by self-fulfilling and self-rewarding expectations. In so doing, the paper sheds light on how impediments to trade and factor mobility can affect the relative importance of history (‘initial endowments’) and expectations (‘initial beliefs’) in determining the evolution of the spatial distribution of economic activities.  2001 Elsevier Science B.V. All rights reserved. Keywords: Agglomeration; Monopolistic competition; Self-fulfilling expectations; Spatial fluctuations; Trade JEL classification: E32; F12; F22; R12

1. Introduction The emergence of economic agglomerations is the subject of a rich body of literature whose origins coincide with economic theory as we know it (see, e.g., *Tel.: 139-2-5836-5450; fax: 139-2-5836-5314. E-mail address: [email protected] (G.I.P. Ottaviano). 0166-0462 / 01 / $ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S0166-0462( 00 )00072-7

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Fujita and Thisse, forthcoming). Among the many related contributions, a recent class of trade models has stressed the role of obstacles to goods and factor mobility in shaping the economic landscape when goods markets are monopolistically competitive.1 These models show that with high barriers to trade and factor mobility, firms’ crowding of local factor and product markets squeezes their operating profits, thus discouraging spatial clustering. However, when barriers to goods and factor mobility are lowered, the negative effect of crowding is relaxed and the expansion of the local market associated with firms’ agglomeration may lead to a more than proportionate expansion of aggregate operating profits (‘market size effect’ or ‘home market effect’, Helpman and Krugman, 1985). This force is used to explain spatial unevenness and to formalize what Scitovsky (1954) calls ‘pecuniary externalities’.2 While such formalization is often regarded as the main contribution of this class of models, it is also at the origin of their major shortcoming. The nonlinearities it induces have so far prevented a thorough forward-looking dynamic investigation so that, while insightful in terms of steady state outcomes, these models are unsurprisingly silent about the adjustment process. As pointed out by Baldwin (2001), there are two sources of intractability. The first is the possibility of multiple steady states, which raises non-trivial issues of global stability analysis and self-fulfilling expectations. The second is the irreducibility of those models to systems of explicit differential equations, which confine the investigation to numerical simulations. The standard ways out are either to appeal to ad hoc dynamic arguments that are not consistent with rational expectations and forward-looking optimizing behavior (e.g., Krugman, 1991b; Krugman, 1992; Krugman and Venables, 1995; Fujita et al., 1999) or to give up pecuniary externalities in favor of more manageable ` 1994). technological spillovers (e.g., Matsuyama, 1991; Krugman, 1991a,c; Galı, Both solutions are unsatisfactory. The problem with the former is logical coherence; the problem with the latter is that the microeconomic origin of macroeconomic externalities is left unexplained. Alternative research strategies have been proposed by Baldwin (2001) on the one side and by Ottaviano (1999) as well as Ottaviano et al. (2001) on the other. Baldwin starts from the observation that, while ad hoc dynamic adjustments 1

A comprehensive presentation of this class of models (sometimes called, with some abuse of name, ‘new economic geography’, Krugman, 1991a) can be found in Fujita et al. (1999) while a handy survey is provided by Ottaviano and Puga (1998). Their relation with work based on either localized technological externalities or spatial competition is discussed by Fujita and Thisse (1996). Their place among the various applications of the monopolistically competitive model of Dixit and Stiglitz (1977) is established by Matsuyama (1995). 2 Scitovsky (1954) distinguishes between ‘technological’ and ‘pecuniary’ externalities. The former materialize through non-market interactions that directly affect the utilities of individuals or the production functions of firms. The latter are by-products of market interactions: they affect individuals and firms only in so far as these are involved in exchanges mediated by the price mechanism.

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remove the first source of intractability, they leave the second untouched so that numerical simulations still have to be used. He then concludes that, if numerical simulations are the cost to be paid for a strict adherence to the original models, then they should be exploited intensively to investigate also the implications of forward-looking behavior. This allows him to introduce full-fledged dynamics in Krugman’s (1991b) core–periphery model. Ottaviano (1999) and Ottaviano et al. (2001) prefer to trade the strict adherence to the original models against the possibility to obtain explicit differential equations.3 In both cases this is achieved by modifying the core–periphery model to get a system of linear differential equations. In the former case this is achieved by restricting the analysis to a specific parameterization of a model with CES demands as the original one. In the latter by replacing the original CES demand system with a linear one. Baldwin (2001) criticizes these linear models because they sacrifice some of the richness of results that make the core–periphery model attractive to theorists. In particular, the linearity makes it impossible for stable interior solutions (‘dispersion’) and stable corner solutions (‘full agglomeration’) to coexist. The aim of this paper is to present a CES model that not only exhibits the same richness of results as the original core periphery model but is also reducible to a set of explicit differential equations. This makes it possible to provide an analytical investigation of the conditions that support the existence of multiple equilibria and endogenous spatial fluctuations driven by self-fulfilling and self-rewarding expectations. In so doing, it sheds light on how impediments to trade and factor mobility can affect the relative importance of history (‘initial endowments’) and expectations (‘initial beliefs’) in determining the evolution of the spatial distribution of economic activities. This paper and Baldwin’s (2001) should be seen as complements. They both ask whether the results in the literature change significantly when ad hoc myopic adjustment is replaced by full-fledged forward-looking dynamics. Following different research strategies they give similar answers. A negative answer when barriers to goods and factor mobility are large as well as agents discount the future heavily. A positive answer when such barriers are low and agents care a lot about the future. In the former case, as under ad hoc myopic adjustment, only history matters. In the latter case, unlike with ad hoc adjustment, expectations play a crucial role. From a technical point of view, as pointed out by Baldwin (2001) the critical issue is the global stability analysis of multiple steady states. Under this respect both research strategies have their own limitations. As it will become clear, this paper provides detailed analytical results when discount rates are small or the state variable is limited to neighborhoods of steady states. Since the core–periphery 3 See also Galı` (1995) for a related effort that studies forward-looking dynamics in a model with local pecuniary externalities. His work however departs quite substantially from the framework surveyed in Fujita et al. (1999) in that he does not allow for trade between locations.

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model can not be reduced to a set of explicit differential equations, Baldwin’s (2001) numerical simulations deal with a more general range of the state variable and large discount factors. However, they provide a gallery of phase portraits, each obtained under a particular set of parameter values. One can not be certain that the gallery is complete. On the one hand, as Krugman (1991b) and Fujita et al. (1999), Baldwin (2001) does not provide any proof that all the steady states of the core–periphery model have been captured by his analysis. On the other, even if all steady state have been captured, he does not prove that all equilibrium trajectories have been detected. The remainder of the paper is organized in five parts. Section 2 presents the model and solves for its short run equilibrium. Section 3 shows that, while amenable to analytical results, it exhibits the same features as the core–periphery model with both myopic and forward-looking expectations. It also states necessary and sufficient conditions on the number of long run equilibria. Section 4 shows that, when the rate of time preference is null, the dynamics of the model are represented by a Hamiltonian system whose perturbation leads to a global stability analysis. Section 5 resorts to a local stability analysis to investigate the case of a discrete rate of time preference. It discusses conditions under which ad hoc myopic dynamics provide an acceptable heuristic device to assess the qualitative properties of forward-looking dynamic adjustment. Section 6 concludes.

2. An analytically solvable core–periphery model This section presents a simple variant of Krugman’s (1991b) core–periphery model that exhibits the same richness of results but, unlike the original model, is amenable to analytical solution.4

2.1. The structure of the model The economy consists of two locations, A and B, with fixed endowments of unskilled labor L and skilled labor H. While unskilled labor is geographically immobile and evenly distributed between locations, each hosting L / 2 unskilled workers, skilled labor can move and its spatial distribution is endogenously determined. Call h(t) [ [0,1] the share of skilled workers in location A at time t. Then, setting H 5 1 by choice of units, h(t) becomes the number of skilled workers in A and 1 2 h(t) the number of skilled workers in B. The instantaneous utility flow is defined over a differentiated (‘modern’) good D and a homogeneous (‘traditional’) good Y with shares a and 1 2 a respectively, a [ (0,1): 4

The model presented appears for the first time in Ottaviano (1996). More recently its static version has been independently proposed by Forslid (1999).

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D (t) Y (t) FS]] D S]] D G, a 12a a

Ui (t) 5 ln

i

55

12 a

i

i 5 A,B

(1)

The differentiated good D is a CES aggregate of domestic and foreign varieties: Di (t) 5

FE

s [n i (t )

d ii (s,t)

s 21 ] s

ds 1

E s [n j (t )

d ji (s,t)

s 21 ] s

G

ds

s ] s 21

(2)

where d ji (s,t) is the consumption in location i at time t of a certain variety s which is produced in j; n i (t) and n j (t) are the ranges of varieties produced respectively in location i and in location j at time t. Without loss of generality, varieties are ordered such that the n’s can be thought of as segments on the real line, so d ij (s,t) is a function also of the variety index s. Finally, s [ (1, 1 `) is the elasticity of substitution between any two varieties and the own-price elasticity of demand for each variety (Dixit and Stiglitz, 1977). The homogeneous good Y is produced in a perfectly competitive sector that uses unskilled labor as the only input. Technology exhibits constant returns to scale with unit input coefficient equal to 1. On the contrary, the differentiated good is produced in a monopolistically competitive sector using both skilled and unskilled labor. Entry and exit are free and take place instantaneously so that equilibrium profits are zero at any time. The production of each variety exhibits increasing returns to scale and technology is represented by a linear cost function with a fixed component g 5 1 undertaken in terms of skilled labor and a marginal component k undertaken in terms of unskilled labor. Because of increasing returns, product differentiation, and free entry, in equilibrium there is a 1:1 relationship between firms and varieties. Moreover, since g 5 1 there is also a 1:1 relationship between firms and skilled workers. Thus, any moment each location produces as many varieties (has as many firms) as the number of skilled workers it hosts: nA (t) 5 h(t) and nB (t) 5 1 2 h(t). Trade in the homogeneous good is free while it is costly in the differentiated good. Trade costs are frictional and are modeled as iceberg costs in the sense of Samuelson (1952): if a unit of the differentiated good has to reach a certain location from the other, t [ [1, 1 `) units must be shipped. In addition, as it is customary in the related literature, any form of intertemporal trade is ruled out ` 1995). (see, also, Matsuyama, 1991; Krugman, 1991c; Galı, Finally, differently from unskilled labor, skilled labor can migrate but, when moving between locations, migrants incur a cost which depends on the rate of ~ ; dh(t) / dt (Mussa, 1978). More precisely, a migrant incurs a migration, h(t) ~ u /g with g [ (0, 1 `) which means that each marginal utility loss equal to uh(t) migrant imposes a negative ‘technological externality’ on other migrants: the larger the number of migrants, the larger the cost of migration. This assumption may be seen as capturing in a simple way the hardships that people face in reality when taking part in large migration flows (Krugman, 1991c).

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It is worthwhile noticing that the assumption that both skilled and unskilled labor are employed in the production of the differentiated good is the only difference with respect to Krugman’s (1991b) model. However, it is enough to achieve the twofold aim of simplifying the analysis without altering the qualitative insights of the original model. In particular, what will turn out to play a key role is the fact that after choosing Y as a numeraire, since unskilled labor is perfectly mobile between sectors in the same location, free trade in the homogeneous good pins down its equilibrium salary to unity in whatever sector it is employed and even if it is geographically immobile. This is true as long as the homogeneous good is produced by both locations. That is the case when a single location alone cannot supply economy wide demand, i.e. when good Y has a large weight in utility (a small) and product variety is highly valued by consumers (s small). The exact condition is a , s /(2s 2 1) and it is assumed to hold from now on.5

2.2. The short-run equilibrium Krugman (1991b) distinguishes between short-run and long-run equilibria. In the former skilled as well as unskilled workers maximize utility and firms maximize profits taking location as given (‘instantaneous equilibrium’). In the latter skilled workers maximize utility also with respect to location so that they have no incentive to move (‘steady state’). This subsection characterizes the instantaneous equilibrium of the model. The next section will be concerned with steady states first under myopic adjustment and then under forward-looking behavior. Given (1), utility maximization gives standard CES demand by residents in location i for a variety produced in location j: pji (s,t)2 s ]]] d ji (s,t) 5 a I (t), i, j 5 hA,Bj qi (t)12 s i

(3)

where qi (t) is the local CES price index associated to (2):

F

qi (t) 5

E

s [n i (t )

pii (s,t)12 s ds 1

E s [n j (t )

pji (s,t)12 s ds

G

1 ] 12 s

(4)

while Ii (t) is local income. Specifically, IA (t) 5 rA (t)h(t) 1 L / 2 and IB (t) 5 rB (t)(1 2 h(t)) 1 L / 2 where r i (t) is skilled labor wage in location i and, as already argued, unskilled labor wage equals 1 everywhere. A typical firm located in location i maximizes profit: 5 This assumption of no specialization – which is a standard trick in trade models – prevents salaries from diverging in the two locations. While admittedly restrictive, it is made here to allow a global stability analysis in a two-dimensional phase plane.

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Pi (s,t) 5 pii (s,t)dii (s,t) 1 pij (s,t)dij (s,t) 2 k[d ii (s,t) 1 t d ij (s,t)] 2 ri (t)

(5)

where again unskilled labor wage is set equal to 1 and t d ij (s,t) represents total supply to the distant location j inclusive of the fraction of product that melts away in transit due to the iceberg costs. The first order condition for maximization implies: pii (s,t) 5 ks /(s 2 1) and

pij (s,t) 5 t ks /(s 2 1)

(6)

for every i and j so that the CES price index (4) simplifies to: 1

] ks 12 s qi (t) 5 ]]fn i (t) 1 r n j (t)g s 21

(7)

where r ; t 12 s [ (0,1] is the ratio of total demand by domestic residents for each foreign variety to their demand for each domestic variety. In general r is less than one and it is equal to one only when trade is free (t 5 1). Thus, since the total number of modern firms is given by n i (t) 1 n j (t) 5 1, (7) decreases (increases) with the number of local (distant) firms. Due to free entry, a firm’s scale of production is such that operating profits exactly match the fixed cost. In other words, its operating profits are entirely absorbed by the wage bill of skilled workers: r i (t) 5 pii (s,t)d ii (s,t) 1 pij (s,t)d ij (s,t) 2 k[d ii (s,t) 1 t d ij (s,t)], that is, given (6), r i (t) 5 kx i (t) /(s 2 1) where x i (t) 5 [d ii (s,t) 1 t d ij (s,t)] is total production by a typical firm in location i. This last expression can be used to determine the output of firms in A and B. Using (3), (6) and (7) as well as n i (t) 5 h(t) and n j (t) 5 1 2 h(t), it implies:

F

G

F

G

F

G F

G

L ks L kxB (t) a ] 1 ]]xA (t) ra ] 1 ]](1 2 h(t)) 2 s 2 1 2 s 21 xA (t) 5 ]]]]]] 1 ]]]]]]]] h(t) 1 r (1 2 h(t)) r h(t) 1 (1 2 h(t)) L ks L kxB (t) ra ] 1 ]]xB (t) a ] 1 ]](1 2 h(t)) 2 s 2 1 2 s 21 xB (t) 5 ]]]]]] 1 ]]]]]]] h(t) 1 r (1 2 h(t)) r h(t) 1 (1 2 h(t))

(8)

(9)

Solving system (8)–(9) for xA (t) and xB (t) yields: xA (t) r h(t) 1 c (1 2 h(t)) ]] 5 ]]]]]] xB (t) c h(t) 1 r (1 2 h(t))

(10)

where the constant c is defined as follows: 1 a c ; ] 1 1 r 2 2s1 2 r 2d] 2 s

F

G

(11)

Eq. (10) shows how the spatial distribution of skilled workers affects their relative wages in the two locations. If r . c [ r , c ], i.e. r . (s 2 a ) /(s 1 a )

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[ r , (s 2 a ) /(s 1 a )], then wages are larger [smaller] in the location which is better endowed with skilled labor. To gain insight on these results, consider the following thought experiment. Start from an initial situation in which skilled workers are evenly distributed between locations and consider what happens when some of them are shifted from one location to the other. On the one hand, for given local expenditures on the modern good, in the location of destination markets become crowded by firms so that their operating profits are squeezed (‘market crowding effect’). On the other hand, because of the inflow of skilled workers, local expenditures increase and this may offset the squeezing effect of market crowding on profits (‘market size’ or ‘home market effect’, Helpman and Krugman, 1985). Intuitively, the latter effect will be stronger than the former if: (i) varieties are bad substitutes (s small) because market crowding has a weak effect on operating profits; (ii) trade costs are small (t small) because firms do not care much about local competition; (iii) the share of expenditures that skilled workers devote to the differentiated good is large (a large) because the impact of immigration on total expenditures is large. As it will become clear in the next section, for the dynamic analysis, it is useful to evaluate the instantaneous indirect utility differential between two skilled workers residing in different locations. Given (1), the indirect utility of a skilled worker in location i is: Wi (t) 5 lnfr i (t) /s qi (t)d ag

(12)

so that:

F

r h(t) 1 c (1 2 h(t)) WA (t) 2 WB (t) 5 ln ]]]]]] c h(t) 1 r (1 2 h(t))

G

F

a h(t) 1 r (1 2 h(t)) 1 ]]ln ]]]]] s 21 r h(t) 1 (1 2 h(t))

G (13)

Hence: Proposition 1. Eq. (13) represents the indirect utility differential as an explicit function of the share of skilled workers in location A. This is the crucial analytical result that can not be obtained in the original core–periphery model and it shows the impact of skilled workers’ migration on their relative instantaneous indirect utilities. Being an explicit function, its properties can be studied analytically without appealing to numerical simulations. To simplify the notation, call f(h) ; WA (t) 2 WB (t) where the dependence of h upon time is left implicit. Eq. (13) consists of two parts. On the right hand side, from left to right, the first logarithmic term measures the welfare value of the wage rate differential (10) that has already been discussed. The second term measures the welfare value of price index differences and it is a monotone increasing function of h: since imported varieties incur the trade cost, the price index is always lower where there are more skilled workers and therefore more varieties are produced locally. The more so (i)

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the worse substitutes goods are (the lower s ) because consumers cannot do without imports; (ii) the larger trade costs are (the larger t ) because many resources have to be devoted to importing; (iii) the larger the modern good share of expenditures (the larger a ) because it is on the modern good that trade costs are paid. As shown in Appendix A, it is straightforward to see that: Corollary 2. The indirect utility differential (13) can take only the three alternative shapes depicted as cases (a),(b), and (c) in Fig. 1.6

Fig. 1. The indirect utility differential.

The corresponding necessary and sufficient conditions are summarized in Table 1 in terms of the two bundling parameters:

s 2a s 2a 21 w ; r 2 ]] ]]] s 1a s 1a 21

(14)

and a ]

r s 21 a F ; r 2 ]] 1 1 r 2 2 (1 2 r 2 )] 2 s 6

F

G

(15)

The same is argued but never proved about the original core–periphery model (see, e.g., Krugman, 1991b; Fujita et al., 1999; Baldwin, 2001).

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Table 1 Necessary and sufficient conditions for Fig. 1

w ,0 w .0

F ,0

F .0

Fig. 1(a) Fig. 1(d)

Fig. 1(c) Fig. 1(b)

Case (a) arises if and only if both w and F are negative; case (b) if both are positive; case (c) if the former is negative while the latter is positive. The remaining case (d) is not feasible because a positive w is not compatible with a negative F. Stated differently, case (a) arises if and only if the elasticity of substitution s is large, if the modern sector expenditure share a is small, and if trade costs t are large so that the market crowding effect dominates the market size effect for every spatial distribution of skilled workers. Case (b) arises when the converse holds. Case (c) arises for intermediate values of the parameters. To summarize, given some spatial distribution h, when (h 2 1 / 2)f(h) . 0 skilled workers’ location decisions to move towards the location with the already larger skilled labor endowment are mutually rewarding or complementary. When this happens, the absolute value of the slope of f(h), say f 9(h), measures the strength of the complementarity. Under this respect, in case (a) skilled workers decision are not complementary. In case (b) they are complementary for any h. In case (c) they are complementary only if h lies outside the two threshold values h L and h H . In other words, for intermediate parameter values, incentives towards agglomeration arise only if the modern sector in one of the two locations has reached some ‘critical mass’.

3. Transitionary dynamics and the long-run This section characterizes the transitionary dynamics and the steady states of the model presented in the foregoing under both myopic and forward-looking expectations.

3.1. Myopic dynamics Krugman (1991b) does not specify any formal migration process. Nonetheless, he assumes that (i) agents are myopic in the sense that they make their location choices on the basis of current payoffs only, and (ii) relocation is a smooth process in the sense that the number of relocating agents is a continuous function of time as long as all agents are not agglomerated in a single location. In the present model this amounts to assuming that at any instant t skilled workers care only about the

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instantaneous indirect utility differential (13) and, if they happen to be in the location that offers the lower indirect utility, they relocate as long as the cost of migrating is offset by the utility gain. Leaving the time dependence of variables implicit, this can be formalized as: f(h) if h [ (0,1) ~h /g 5 minh0, f(h)j if h 5 1 maxh0, f(h)j if h 5 0

5

(16)

where h~ /g is the migration cost assumed above.7 If f(h) is positive, some skilled workers will move from B to A; if it is negative, some will go in opposite direction. Clearly, a steady state implies h~ 5 0, which is the case for an interior point h [ (0,1) whenever f(h) 5 0 and at a boundary h 5 1 (h 5 0) whenever f(h) . 0 ( f(h) , 0). Moreover, a steady state is stable for (16) if, for any marginal deviation, this equation of motion brings the distribution of skilled workers back to the original one. Therefore, the agglomerated configuration is always stable when it is a steady state while the dispersed configuration is stable if and only if f 9(h) is nonpositive in a neighborhood of this point. We can use Table 1 to assess the number and the stability of steady states. In case (a), which arises when both w and F are negative, only an interior steady state exists at h 5 0.5 (‘dispersion’) and it is stable. In case (b), which arises when both w and F are positive, three steady states exist at h 5 0.5 as well as at h 5 0 and h 5 1 (‘full agglomeration’). All three are stable. In case (c), which arises when w is negative while F is positive, five equilibria exist. As in case (b), three correspond to h 5 0.5, h 5 0, h 5 1, and are stable. The remaining two are symmetrically situated around h 5 0.5 (‘partial agglomeration’) and are unstable. Thus, the condition w 5 0 implicitly defines what Fujita et al. (1999) call the break point, that is, the value of trade costs below which the steady state at h 5 0.5 becomes unstable. The condition F 5 0 implicitly defines what they call the sustain point, that is, the value of trade costs below which h 5 0 and h 5 1 correspond to stable steady states. As proved in the Appendix A, the break point is always smaller than the sustain point so that, as long as trade costs fall between those two values, dispersion and full agglomeration are both stable steady state outcomes. All this shows that: Proposition 3. With myopic behavior the present model mimics the crucial properties of the original core–periphery model by Krugman (1991b). 7

Baldwin (2001) assumes instead a migration cost equal to h~ / [g h(1 2 h)] on the ground that such choice formalizes the adjustment process in Fujita et al. (1999). The present analysis prefers to stick to the original approach in Krugman (1991b). Incidentally, notice that the term h(1 2 h) introduces additional externalities in the migration decision.

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3.2. Forward-looking dynamics When multiple stable steady exist, the myopic adjustment (16) gives a clear-cut answer to the question of what selects which stable steady state is approached in the long run. The answer is history (‘initial endowments’), that is, the initial distribution of skilled workers. In case (b), if initially h happens to be larger (smaller) than 0.5, the economy eventually stops at h 5 1 (h 5 0). In case (c), it moves towards dispersion (full agglomeration) if initially h happens to be inside (outside) h L and h H . The aim of the rest of the paper is to check whether this answer still holds when skilled workers make forward-looking migration decisions having perfect foresight about the future paths of indirect utilities in the two locations hWA (t),WB (t)j `0 . Consider a generic instant t and let vA (t) [vB (t)] be the discounted sum of the current and future instantaneous indirect utilities of a skilled worker who resides in A [B] from t onwards: T

E

vA (t) 5 WA (s) e 2d (s 2t ) ds 1 vA (T ) e 2d (T 2t )

(17)

t T

E

vB (t) 5 WB (s) e 2d (s 2t ) ds 1 vB (T ) e 2d (T 2t )

(18)

t

where d [ [0, 1 `) is the rate of time preference and T is the first time the economy reaches a steady state. T can be null, finite or infinite depending on the initial conditions and the type of steady state the system is heading towards.8 At any instant t, along an equilibrium trajectory each skilled worker must be indifferent between moving and staying put. For this to be the case, it must be that at any t the migration cost just offsets the discounted sum of the current and future indirect utility differentials:

~ /g 5 vA (t) 2 vB (t) h(t)

(19)

In addition, at any instant t, along an equilibrium trajectory each skilled worker must be indifferent between moving and postponing migration to the very last ~ ) 5 0) instant T. Since at T the economy reaches a steady state, motion stops (h(T so that the migration cost is null. Thus, for a skilled worker to be indifferent between moving at t and waiting for T, it must be that the two alternatives give the same payoff: 8 As it will be shown, there are only three relevant cases. T will be null if the economy starts at a steady state; it will be infinite if the economy heads towards a saddle point; it will be finite if the economy is going to rest at a boundary.

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~ /g 5 [vA (T ) 2 vB (T )] e 2d (T 2t ) vA (t) 2 vB (t) 2 h(t)

63

(20)

Before proceeding, it is convenient to define the following shadow price: v(t) ; vA (t) 2 vB (t)

(21)

which represents the ‘private’ value for a skilled worker of residing in location A rather than in location B. We are now ready to characterize the transitionary dynamics when skilled workers are forward-looking. First, given (17) and (18), differentiating v(t) with respect to t and leaving the dependence of variables on t implicit give: v~ 5 d v 2 f(h)

(22)

where v~ ; dv(t) / dt. Second, given (21), (19) can be rewritten as: h~ 5 g v

(23)

Third, given (19), (20) implies v(T ) 5 0 (Fukao and Benabou, 1993). Thus, while with myopic behavior the dynamics of migration are represented by the differential equation (16), with forward-looking behavior they are represented by the twodimensional system of differential equations (22, 23). In particular, (23) shows that some skilled workers will move from B to A if v is positive, while some will go in opposite direction if v is negative. Since v 5 et` f [h(s)] e 2d (s 2t ) ds, this reveals the importance of the future for forward-looking agents. In steady state motion stops so that h~ 5 0 and v~ 5 0. This happens for internal values of h such that f(h) 5 0 and for terminal values h 5 0 and h 5 1 because of the boundary condition v(T ) 5 0. What is notable is that the steady state values of h do not depend on the values of d and g so that: Proposition 4. With forward-looking behavior the steady states of the model are the same as with myopia. Thus, Table 1 can be used again to establish under which conditions the system will exhibit unique or multiple equilibria. Let us turn now to the study of the stability of steady states. As far as possible this issues will be addressed analytically by supplementing the drawing of phase diagrams by approximations along two dimensions. Section 4 will assess the global stability of the model for values of d in the neighborhood of zero. Section 5 will investigate its local stability properties in a neighborhood of steady states for

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discrete d. The reason for having a global analysis together with a local one comes from the nonlinearity of system (22)–(23).

4. Global stability analysis Solving the system (22)–(23) is not an easy task because f(h) is nonlinear and therefore precludes an exact analytical solution. That is why any analytical result has to rely on approximations. The standard practice is to restrict the attention to linear approximations around the steady states. For nonlinear systems this is unsatisfactory for the study not only of global but also of local stability (Grandmont, 1985). I try to overcome this problem by supplementing the linear analysis by an approximated global stability analysis based on parameter perturbation (asymptotic) methods. ‘‘Generally, in perturbation methods one starts with an (integrable) system whose solutions are known completely and studies small perturbations of it. Since the unperturbed and perturbed vector fields are close, one might expect that solutions will also be close [ . . . ] for finite times’’ (Guckenheimer and Holmes, 1990).9 This section shows that, when the rate of time preference is zero (d 5 0), the system (22)–(23) is a solvable Hamiltonian system, that is, there exists a function H(h,v) (the Hamiltonian) whose level curves give the global structure of the phase ~ 2 ≠H / ≠h). By perturbing the Hamiltonian diagram (formally, h~ 5 ≠H / ≠v and v5 system, the global structure of the phase diagram can then be characterized to whatever degree of approximation for a positive infinitesimal d. Even if unrealistic, the case of infinitesimally small discounting is the natural benchmark to be compared with the myopic case of complete discounting. It is worthwhile to clarify the difference between the perturbation method and the simulation approach adopted by Baldwin (2001). Since an analytical expression for the instantaneous indirect utility differential is out of reach in the original core–periphery model, given alternative sets of parameter values Baldwin first approximates the numerical expressions of such differential and then plots the phase diagrams for those particular approximations. Thus, what he provides is a gallery of phase portraits, each obtained under a particular set of parameter values. One can not be certain that the gallery is complete. On the one hand, Krugman (1991b), Fujita et al. (1999) as well as Baldwin (2001) do not provide any proof that all the steady states of the core–periphery model have been captured by their respective analyses. On the other, even if all steady state have been captured, in Baldwin (2001) there is no proof that all equilibrium trajectories have been identified. 9 This method is borrowed from physics and has been proposed for the first time in economics by Matsuyama (1991). The parallel between the present analysis and a corresponding problem in physics is discussed in Ottaviano (1996).

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4.1. Hamiltonian representation Assume d 5 0. Then, the Hamiltonian associated with (22)–(23) is

g H(h,v) 5 ]v 2 1 F(h) 2

(24)

where, by definition,

E

1 F(h) 5 f(h) dh 5 ]]h[ r h 1 c (1 2 h)]ln[ r h 1 c (1 2 h)] r 2c 1 [c h 1 r (1 2 h)]ln[c h 1 r (1 2 h)] 2 r 2 c j a 1 ]]]]h[h 1 r (1 2 h)]ln[h 1 r (1 2 h)] (s 2 1)(1 2 r ) 1 [ r h 1 (1 2 h)]ln[ r h 1 (1 2 h)] 2 1 2 r j This statement is readily proved by noticing that for d 5 0, given (24), v~ 5 2 ≠H / ≠h and h~ 5 ≠H / ≠v yield (22) and (23) respectively. Moreover, dH / ~ ~ 2 vh ~ ~ 5 0, where the second equality is granted by (22) dt 5 (g v)v~ 1 f(h)h~ 5 hv and (23). This implies that the level curves of the Hamiltonian provide the phase portrait of the system (22)–(23). Then, for a generic level u, (24) can be solved for the equilibrium trajectories: ]]]] 2(u 2 F(h)) v 5 6 ]]]] g

œ

(25)

At which level u the system actually moves is determined by the initial conditions, that is, the initial distribution of skilled workers between locations, h(0), and the sum of current and expected future instantaneous indirect utility differentials, v(0). The evaluation of such sum depends on the steady state that is expected to be reached. Indeed, when the appropriate conditions on parameter values in Table 1 are met, for a given h(0) multiple steady states can be perfectly forecast giving rise to multiple v(0) and multiple levels u. Thus, even if the economy starts with a given h(0), multiple transitional paths are possible, one for each level u, and a steady state will be eventually reached just because it is the one that is expected to emerge (‘self-fulfilling prophecy’). Hence: Proposition 5. With forward-looking behavior, when the rate of time preference is zero, history is irrelevant and expectations are all that matters for the selection among multiple steady states. This is exactly the opposite of what happens when agents are myopic and discount the future completely. A typical property of Hamiltonian systems is that all interior steady states are either centers or saddle points. This comes from the fact that the trace of the

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Jacobian matrix of the associated system of differential equations is zero at an interior steady state. Specifically, for a given steady state (h,0), the Jacobian matrix associated to system (22)–(23) is: Jh ;

F

0 g 2 f 9(h) d

G

(26)

with trace trJh 5 d. Then, with no discounting the trace evaluates to zero, which means that, in the neighborhood of the steady state, the amplitude of the motion is constant through time. In correspondence to a saddle point the Hamiltonian has a saddle. In correspondence to a centre it has a maximum or a minimum. While at a saddle point there is the meeting of four branches of a level curve, in the neighborhood of a centre level curves are non-overlapping closed trajectories or orbits. The motions corresponding to the closed orbits are periodic and, because F(h) is nonlinear, the period is a function of the amplitude of the motion. Since the amplitude depends on which level curve the system moves, the period is ultimately a function of initial conditions. Figs. 2–4 illustrate the global stability properties of the system as they can be read through the Hamiltonian under the three possible shapes of f(h) associated to cases (a), (b) and (c). In the upper portion of each picture, the undulating line represents F(h), that is, the section of the Hamiltonian at v 5 0. Since the sign of the Hessian of H(h,v) is sign[(≠ 2 H / ≠v 2 )(≠ 2 F(h) / ≠h 2 )] 5 sign[ f 9(h)], this section summarizes all the relevant information on the way all the parameters (other than g ) influence the shape of H(h,v). The straight dotted horizontal lines represent alternative values of u and by (24) the vertical distance between a given horizontal line and the undulating line represents g v 2 / 2. For a given u motion takes place along the corresponding level curve of the Hamiltonian. This is shown in the lower portion of each figure, which portrays the phase diagram plotting v 5 h~ /g against h. The dark solid curves represent the level curves of the Hamiltonian passing through the steady states, while the light arrows show the direction of the vector field. Fig. 2 represents case (a), in which F(h) has a maximum at h 5 0.5 that corresponds to a saddle point. Since this is the only steady state, neither history nor expectations determine the long run distribution of skilled workers. The opposite case is depicted in Fig. 3 that refers to case (b). Since F(h) has a minimum at h 5 0.5 corresponding to a centre, there are endogenous cycles of any period and any amplitude. Two properties of the model explain the existence of closed trajectories: the complementarity between skilled workers’ location decisions and zero discounting. Finally, Fig. 4 represents case (c). F(h) has now a maximum in h 5 0.5 and two minima in h 5 h L and h 5 h H , which correspond respectively to a saddle point and two centres. Accordingly, one observes a tendency towards the dispersed equilib-

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Fig. 2. The Hamiltonian: A unique saddle point.

rium point (0.5,0), oscillations between (very uneven) spatial distributions of firms, and oscillations around either (h L ,0) or (h H ,0). Fig. 4 shows in a somewhat dramatic way that in nonlinear systems local and global stability can be very different.

4.2. Perturbation of the Hamiltonian The previous section has shown that, in the presence of multiple steady states, zero discounting (d 5 0) makes history irrelevant and only expectations matter. Thus, under forward-looking behavior the implications of myopic adjustment are

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Fig. 3. The Hamiltonian: A unique centre.

completely reversed. By perturbing the Hamiltonian system we now assess whether such result is modified for a positive infinitesimal d. When d . 0 the trace of the Jakobian (26) is positive, which causes the amplitude of the motion to increase through time. The effects of turning from d 5 0 to d . 0 are shown in Fig. 5. The dark solid curves represent the trajectories that solve the system (22)–(23). There are no analytical expressions for such curves. However they can be approximated at whatever degree of precision (Nayfeh, 1973; Nayfeh and Mook, 1979). The light arrows show the direction of the vector field. Cases (a), (b), and (c) correspond respectively to Figs. 2, 3 and 4. In case (a) there is a unique steady state with dispersion and it is a saddle point.

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Fig. 4. The Hamiltonian: One saddle and two centres.

Perturbation makes no qualitative difference since even in the Hamiltonian case there are no closed orbits. More interesting is case (b) where perturbation breaks the closed orbits around the dispersed steady state (0.5,0). Since the amplitude of the motion increases through time, the dispersed steady state is now a source and the system will eventually reach an endpoint, either (0,0) or (1,0), in finite time. Initial values for h can be divided into two groups. In correspondence of some of them, relatively close to h 5 0.5, there are two trajectories leading towards full agglomeration in alternative locations. For such initial values, expectations decide along which trajectory the economy will move. The set of those values is what Krugman (1991c) calls the ‘overlap’. On the contrary, for other initial values of h, relatively

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Fig. 5. Perturbation of the Hamiltonian.

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close to either 0 or 1, there is a unique trajectory so that expectations have no role to play, and the initial value of h alone determines the final outcome. If initially a location happens to have a much lower share of skilled labor, it will eventually lose all its modern sector because of full agglomeration in the other location. Finally, in case (c) agglomeration and dispersion coexist as stable steady states and the corresponding dynamics can be very rich with multiple overlaps centered around (0.5,0), (h L ,0) and (h H ,0). In particular, the unraveling of the Hamiltonian system for a positive infinitesimal d reveals the existence of trajectories that, after leaving (h L ,0) [(h H ,0)], spiral around both (h L ,0) and (h H ,0) before hitting full agglomeration. While similar trajectories do not appear in Baldwin (2001), they are likely to exist also in the original core–periphery model with forward-looking expectations. As in the case of the other trajectories identified by Baldwin (2001, Fig. 7), their existence means that the economy could jump from the dispersed steady state onto a path leading to full agglomeration just because all skilled workers expect this to happen. Once more, this reveals the limits of history and the relevance of expectations when agents care a lot about the future. To summarize: Proposition 6. With forward-looking behavior, when the rate of time preference is infinitesimally small, either history or expectations matter for the selection among multiple steady states depending on parameter values and initial conditions.

5. Local stability analysis With null or infinitesimal discounting endogenous spatial fluctuations are always associated with multiple equilibria. This section investigates the generality of this feature by studying the case of discrete positive discounting. Since in this case the deviation of d from zero is not small, the perturbation tool has to be abandoned. This has a cost: because the stability analysis can be performed only in the neighborhood of steady states, global stability results are precluded. ‘Overlaps’ exist as in Fig. 5(b,c) only if the solution of the system yields multiple whirling trajectories. Since the demarcation curves, h~ 5 0 and v~ 5 0, are everywhere differentiable, this happens whenever the Jakobian matrix (26) has complex eigenvalues, that is, whenever (trJh )2 , 4uJh u, where trJh and uJh u are respectively its trace and determinant. Given (26), this condition can be rewritten as:

d 2 2 4gf 9(h) , 0

(27)

Hence: Proposition 7. With forward-looking behavior, when the rate of time preference is

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discrete, expectations matter for the selection among multiple steady states if (27) holds in the neighborhood of at least one steady state. In other words, if (27) holds, endogenous fluctuations appear together with multiple steady states so that, whenever the initial h lies inside the ‘overlaps’, expectations can overturn history. On the contrary, if (27) is violated, the existence of only real eigenvalues rules out the possibility of whirling trajectories. Multiple steady states do not imply endogenous spatial fluctuations and the initial endowments select the steady state towards which the economy moves. Thus: Corollary 8. With forward-looking behavior, if (27) does not hold for any steady state, then current instantaneous indirect utility differentials can be used as a heuristic device to assess the direction of migration flows as with myopia. Condition (27) holds for relatively large positive f 9(h), small d and large g, i.e. if the complementarity between location decisions is strong, skilled workers are patient, and the costs of migration are small so that the adjustment is fast. It clarifies that, differently from the case of infinitesimal discounting, with a positive rate of time preference the possibility of self-fulfilling expectations depends not only on the existence of complementarity, as revealed by a positive f 9(h), but also its intensity, as measured by the absolute value of f 9(h). This is larger for larger a and smaller s, while it is non-monotone in t. For example, in the particular case of h 5 0.5, it can be shown that f 9(0.5) reaches a global maximum for some value of r in between [(s 2 a )(s 2 a 2 1)] / [(s 1 a )(s 1 a 2 1)] and 1. Starting with autarchy, as trade barriers are gradually lowered, at some point expectations start to affect the spatial distribution of economic activities. However, as barriers get lower and lower, the relevance of expectations first weakens and then disappears. The economic intuition for these results may be explained as follows. For concreteness, take Fig. 5(b), suppose an initial distribution of skilled workers at h , 1 / 2 and ask what is needed to reverse an ongoing agglomeration process currently leading towards h 5 0. If the evolution of the economy were to change direction, skilled workers would experience falling instantaneous indirect utility flows for some time period as long as h , 1 / 2. The instantaneous indirect utility flows would start growing only after h becomes larger than 1 / 2. Accordingly, skilled workers would first experience utility losses followed later by utility gains. Since the losses would come before the gains, they would be less discounted. This provides the root for the intuition behind the results. When the complementarity leads to substantial wage rises (that is, for intermediate values of t ), the benefits of agglomerating at h 5 1 can compensate workers for the losses they incur during the transition phase, thus making the reversal of migration possible. On the contrary, when these externalities get weaker (that is, for low or high values of t ), the benefits of agglomerating at h 5 1 do not compensate workers for the losses.

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As a consequence, the reversal in the migration process may occur only for intermediate values of t. As to the remaining comparative static properties, they are explained by fact that a large rate of time preference gives more weight to the utility losses, and a slow speed of adjustment extends the time period over which skilled workers’ wellbeing is reduced. Fig. 6 summarizes the results of the local stability analysis for different values of the trade cost parameter t and given a, s, g and r such that (27) holds for some t. To understand the picture, let t increase gradually from 1. Initially, (h,v) 5 (0.5,0) is the only interior steady state and it is an unstable node (i.e., there are no whirling trajectories) until t reaches t 1 . At this point (0.5,0) becomes an unstable focus (i.e., there are whirling trajectories) so that, as long as t remains below t 2 , trajectories diverge in spirals. However, as t rises above t 2 the spirals disappear and (0.5,0) is again an unstable node. At tB (the break point defined by w 5 0) the steady state (0.5,0) becomes a saddle point and two other interior steady states (h L ,0) and (h H ,0) appear.10 They are unstable nodes as long as t stays below t 3 . Above this threshold they become unstable foci until t reaches tS (the sustain point defined by F 5 0) for which value h L and h H hit the boundaries 0 and 1 respectively. Therefore: Corollary 9. Assume that (27) holds for some t. Then, with forward-looking behavior and a discrete rate of time preference, multiple equilibria are associated with endogenous fluctuations if t belongs to [t 1 ,t 2 ] or [t 3 ,t S ].

Fig. 6. The bifurcation diagram. 10

This is an example of a reverse subcritical pitchfork bifurcation (Guckenheimer and Holmes, 1990).

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It can be shown that the width of the two intervals increases with d and decreases with g. In particular, t 1 goes to 1 while t 2 and t 3 go to tB from opposite sides as d goes to zero which is consistent with the analysis of the previous sections (see Ottaviano et al., 2001, for similar results in a linear model).

6. Final remarks Recently, a growing number of models have tried to explain the emergence of economic agglomerations through Dixit-Stiglitz monopolistic competition in the presence of obstacles to trade and factor mobility. Most of these models have addressed what is an inherently dynamic issue via an essentially static approach. A thorough exploration of their dynamic implications had been so far postponed due to two main reasons. First, existing models typically exhibit multiple steady states, which raises delicate issues of global stability analysis and self-fulfilling expectations. Second, they can not be reduced to explicit differential equations, which restricts the investigation to numerical simulations. In order to overcome some of these difficulties, this paper has proposed an alternative set-up that, while preserving the attractive features of Krugman’s (1991b) core–periphery model, is nonetheless reducible to a system of explicit differential equations. By supplementing local linearization techniques with global perturbation methods, it has been shown that expectations (‘initial beliefs’) as opposed to history (‘initial endowments’) matter for the long-run spatial distribution of economic activities if the complementarity between agents’ location decisions are strong, if the transition is fast, and if agents are patient. Complementarity is strong if the monopolistically competitive products are bad substitutes for each other and absorb a large fraction of consumers’ expenditures. More interestingly, the strength of the complementarity is a non-monotone function of the level of trade costs and reaches a maximum for intermediate values. Reducing trade barriers from autarchy, at some point expectations start to affect the organization of the economic space. However, as the economy gets more and more closely integrated, their relevance first weakens and then vanishes altogether.

Acknowledgements I am indebted to Richard Baldwin, Jean Gabszewicz, Piero Gottardi, JeanMichel Grandmont, Philippe Monfort, Konrad Stahl, Jacques-Franc¸ois Thisse, Tony Venables, and two anonymous referees for helpful comments. Financial support by the European Commission is gratefully acknowledged.

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Appendix A This appendix studies the properties of the indirect utility differential (13) as a function of h. First of all, even if it is hard to solve for its zeroes analytically, one can readily spot one of them and assess the number of the others. Due to the symmetry of the model, f(h) 5 0 for h 5 0.5. Moreover, since the sign of f 9(h) depends on the sign of its quadratic numerator, f(h) changes slope at most twice. That is, either h 5 0.5 is the only zero of f(h) or there are (no more and no less than) two other zeroes, say h L and h H , that are symmetric around it. The four possible alternative shapes of f(h) are depicted in Fig. 1. In cases (a) and (c), f 9(0.5) , 0, while in (b) and (d), f 9(0.5) . 0. By simple algebra, one gets: 1 2 r (s 1 a )r 2 (s 2 a ) 12r a f 9(0.5) 5 4]] ]]]]]] 1 4]] ]] 1 1 r (s 1 a )r 1 (s 2 a ) 11r s 21

(A.1)

The first term on the right hand side shows the impact of a marginal relocation of skilled workers on relative wages while the second term expresses its impact on relative price indices. While the latter is always positive, the sign of the former depends on parameter values. For notational convenience, define:

s 2a s 2a 21 w ; r 2 ]] ]]] s 1a s 1a 21

(A.2)

which is definition (14) in the main text. It can be easily checked that f(h) is decreasing (increasing) at h 5 0.5 if and only if w , ( . )0. This happens for large (small) t, large (small) s and small (large) a. In the special case where a . s 2 1, w is always positive (a ‘black hole’ situation according to Fujita et al., 1999, which is traditionally ruled out). The above conditions on w help to discriminate between situations like (a) and (c) on the one side and (b) and (d) on the other. However, in order to distinguish between (a) and (c) or between (b) and (d), more information is needed. In cases (a) and (d), f(0) . 0 and f(1) , 0. The opposite is true in (b) and (c). Then, the sign of f(0) and f(1) can be used to distinguish between (a) and (d) on the one side and (b) and (c) on the other. It is readily assessed that f(0) 5 2 f(1). In particular:

F

G

2sr a f(1) 5 2 f(0) 5 ln ]]]]]] 2 ]]ln r 2 s 2 1 (s 1 a )r 1 (s 2 a )

(A.3)

This expression measures the instantaneous indirect utility differential when the modern sector is agglomerated in a single location. The two terms on the right hand side show how much of this differential stems from different wages and different price indices respectively. For notational convenience, define:

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r s 21 a F ; r 2 ]] 1 1 r 2 2 (1 2 r 2 )] 2 s

F

G

(A.4)

which is definition (15) in the main text. It is easily verified that f(h) behaves as in (a) and (d) [(b) and (c)] if and only if F , ( . )0. Moreover, in its dependence upon parameters f(1) exhibits the same qualitative behavior as f 9(0.5). Since f(0.5) is always null and f 9(h) changes sign at most twice, by crossing the conditions on F and w, one gets necessary and sufficient conditions for each picture to describe the actual behavior of f(h). These conditions are summarized in Table 1. Case (d) has to be ruled out. The reason is that F , 0 implies w , 0, while the reverse is not true.

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