Econ Theory (2013) 53:213–238 DOI 10.1007/s00199-011-0686-7 RESEARCH ARTICLE
Spatial inequality, globalization, and footloose capital Toshiaki Takahashi · Hajime Takatsuka · Dao-Zhi Zeng
Received: 19 October 2010 / Accepted: 12 December 2011 / Published online: 27 December 2011 © Springer-Verlag 2011
Abstract This paper shows the equivalence of spatial inequalities in industrial location and in income by revisiting the home market effect (HME) without any homogeneous good based on a reconstructed footloose capital model. In this simple framework, spatial inequalities in industrial location and in income are the HMEs in terms of firm share and wage, respectively. We show that the larger country has a more-than-proportionate share of firms and a higher wage. Furthermore, both the wage differential and the industrial location in the larger country evolve in an inverted
We wish to thank two anonymous referees, editor Tim Kehoe, Richard Baldwin, Masahisa Fujita, Taiji Furusawa, Yasuhiro Sato and Jen-Te Yao for their highly valuable and detailed comments. Thanks are also extended to the participants at the 6th Annual Meeting of the Asia Pacific Trade Seminars in Osaka, the 10th SAET Conference on Current Trends in Economics in Singapore, the 1st Asian Seminar in Regional Science, the 57th Annual North American Meetings of the Regional Science Association International, and seminars in Renmin University of China and Graduate University of Chinese Academy of Sciences. Financial support from the Japanese Ministry of Education, Culture, Sports, Science, and Technology through Grants-in-Aid for Science Research 20730183, 22330073 for the second author, 22330073, 21243021, 21510147, and the Y. C. Tang disciplinary development fund of Zhejiang University are acknowledged for the third author. T. Takahashi Graduate School of Public Policy, The University of Tokyo, Tokyo 113-0033, Japan e-mail:
[email protected] H. Takatsuka Graduate School of Management, Kagawa University, Takamatsu 760-8523, Japan e-mail:
[email protected] D.-Z. Zeng (B) Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan e-mail:
[email protected] D.-Z. Zeng Center for Research of Private Economy, Zhejiang University, Zhejiang 310027, China
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U-pattern when transport costs decline. Finally, we analytically examine the effects of trade liberalization on the welfare and show that both countries may gain from globalization. Keywords capital
Spatial inequality · Globalization · Home market effect · Footloose
JEL Classification
F12 · R12
1 Introduction In this paper, we address spatial inequalities in industrial location and income, which are important issues both for scholars and policy makers. Since globalization is a major force in world development today, new trade theory (NTT) and new economic geography (NEG) are developed to clarify how globalization impacts on geographic concentration of economic activities and wage inequality. A useful concept in such studies is the home market effect (HME), which refers to the tendency that when manufactured (differentiated) goods produced under increasing returns to scale (IRS) incur transport costs, firms producing these goods concentrate in large countries in order to save transport costs. While many researchers recognize the pervasive HME (Head et al. 2002) as a notable building block of NTT and NEG models, the theoretical base is still under construction. In particular, two HME definitions, the HME in terms of firm share and the HME in terms of wage, are known in the literature (e.g., Behrens et al. 2009, footnote 1). However, the relationship between these two definitions is never explored. More specifically, these two definitions correspond to the spatial inequalities in industrial location and in income, respectively. The HME in terms of firm share, which is more popular, is defined as a phenomenon in which a country with a relatively larger local demand attracts a more-than-proportionate share of manufacturing firms, and thus, the country becomes a net exporter of the good (Krugman 1980, Section III; Helpman and Krugman 1985, Section 10.4). Meanwhile, the HME in terms of wage is defined as the fact that, other things being equal, the wage is higher in a larger country (Krugman 1991, p. 491). In this paper, we intend to show the equivalence of these two definitions and, therefore, the equivalent relationship between the spatial inequalities in industrial location and income. Originally, as the pioneer of HME research, Krugman (1980) established two general equilibrium frameworks. In Section II of Krugman (1980), the author considers a world of two countries with one manufacturing sector (consists of a single industry) producing the differentiated goods. Labor is the only production factor. In this way, the author obtained the HME in terms of wage but failed at observing the HME in terms of firm share. This is because the trade balance limits the reaction of production to demand and makes the industrial location proportional to the country sizes. This “very strong structure” (Krugman 1980, p. 954) removes the impact of transport costs on the industrial location. On the other hand, Section III of Krugman (1980) assumes two
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symmetric industries in the manufacturing sector and two groups of consumers, each consuming differentiated goods in one industry only. The country sizes are equal, but each has a larger share of one consumer group. This framework displays the HME in terms of firm share but not wage. In addition, the analytical solvability of this framework depends heavily on the symmetry between industries and between population group sizes. To allow for the reaction of production to demand in the IRS sector, many authors add one more (agricultural) sector (e.g., Helpman and Krugman 1985), characterized by constant returns to scale (CRS), perfect competition, homogeneity of goods, and costless trade. Since labor is also the only input of agricultural production, the wages (factor prices) in two countries are equalized. In this way, we can observe the HME in terms of firm share but fail at obtaining the HME in terms of wage. In other words, this setup elegantly illustrates the spatial inequality in industrial location; however, it does not demonstrate a spatial inequality in income. Although the general equilibrium analysis is quite simplified by the free-traded agricultural good, this assumption is criticized by some authors (e.g., Davis 1998) because it appears to be at odds with reality. Crozet and Trionfetti (2008) call the agricultural good an “outside good,” suggesting an investigation without it. In the models above, labor is assumed to be the only production factor. However, capital is another production factor widely employed in trade theory. In most industrialized countries, labor mobility is subject to tight restrictions,1 while capital mobility is predominant among countries (Amiti 1998, p. 233). Especially, after the collapse of the Bretton Woods system in the early 1970s, international capital mobility rose rapidly (e.g., Brakman et al. 2006, Ch.6). In addition, foreign direct investment (FDI) inflows also rose over the last 30 years, and the FDI inward stock in developed economies increased significantly from $392 billion in 1980 to $7,117 billion in 2005 (UNCTAD 2003, 2006). Furthermore, Krugman and Venables (1995, p. 876) argue that adding capital movements to models is indeed important for political debate over integration. We can reach our goal by a framework with mobile capital as a production factor but without the agricultural sector.2 This allows us to observe spatial inequalities in industrial location and income simultaneously. Because of the two factors, mobile capital and immobile labor, this model can be interpreted as a modified footloose capital model of Martin and Rogers (1995) by eliminating the agricultural sector or a modified one-industry model of Krugman (1980, Section II) with one more factor of mobile capital. The mobile capital can make firms concentrate in the larger country and brings a higher wage there, exhibiting the HME in terms of both wage and firm share. Intuitively, more firms concentrate in the larger country to save transport costs, which results in a higher wage. Each firm there inputs less labor because of the higher wage. The larger country then ends up with a more-than-proportionate share of firms. The second contribution made in this paper is the analytical proof of the inverted U-pattern of spatial inequalities in both industrial location and income. 1 Puga (1999) finds that the labor immobility is a crucial assumption in such studies. 2 Takatsuka and Zeng (forthcoming) show that mobile capital is essential for the HME by a footloose
capital model with agricultural transport costs. However, they focus only on the HME in terms of firm share and do not clarify the case without the agricultural sector.
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The inverted U-pattern refers to a non-monotonic relationship between the level of trade costs and spatial inequalities. Specifically, starting from high levels of trade costs, a reduction in these initially results in larger inequalities (agglomeration), but further reduction brings smaller inequalities again (redispersion). Inequality is a sensitive issue because some politicians and economists of small regions or countries worry about whether their economies could be permanently de-industrialized through increased economic integration. Such an inverted U-pattern is empirically supported by Williamson (1965), Brülhart and Torstensson (1996), and Barro (2000). While regional inequality is naturally understood as income inequality, most existing theoretical works are based on firm location (e.g., Krugman and Venables 1990, 1995; Venables 1996; Puga and Venables 1997; Zeng and Kikuchi 2009; Zeng and Zhao 2010). There are also some theoretical studies on the income inequality among different social groups within a country [e.g., Kurokawa (2011) for the income inequality between high-skilled and low-skilled workers, and Alexopoulos and Cavalcanti (2010) for the relation between education and income distribution]; however, the income inequality within a group among different countries is not sufficiently explored in the literature. Interestingly, our finding suggests that the U-shaped evolution of spatial inequality is pervasive because it is generated only from the mobility of capital. It is noteworthy that the absence of the agricultural sector releases us from the consideration of agricultural transport costs, which is controversial among many researchers. In fact, all existing studies of U-shaped evolution essentially depend on the costlessly traded agricultural good, which remains contentious as in the previous HME arguments. Moreover, differing from the catastrophic changes in core-periphery models with linkages among firms (Krugman and Venables 1995; Venables 1996; Puga and Venables 1997), we obtain a process of gradual change. The third result is the welfare analysis. We first show that the welfare in the larger country is higher than the welfare in the smaller country, which is also true in traditional models under the assumption of factor-price equalization. The simple model here allows us to go further and examine whether two countries gain from trade liberalization. The relocation of firms increases the price index in the origin market and decreases the price index in the destination for fixed trade costs. However, trade liberalization lowers the trade costs and makes the imported varieties cheaper. Thus, measuring the total effect of decreasing trade costs on the welfare is a formidable task. Helpman and Krugman (1985, p. 179) declare that it is difficult to prove in general that countries gain from trade in the differentiated products model, although their arguments are limited to the cases of free trade and autarky. To the best of our knowledge, there are no analytical studies about the effect of decreasing trade costs on welfare. Fortunately, our simple setup allows us to examine the two opposite impacts of trade liberalization as a whole, and we are able to show how large and small countries have different interest in economic globalization. Specifically, the welfare in the larger country increases when the industry concentrates there, and the welfare in the smaller country increases during its reindustrialization process. Furthermore, both countries gain from trade integration when transport costs are either small or high. The rest part of this paper is organized as follows. In Sect. 2, the model for the HME is established. Sect. 3 examines the equilibrium, and Sect. 4, the welfare. Finally, Sect. 5 summarizes the conclusions.
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2 The model The economy space consists of two countries (or regions) j = 1, 2. The countries have the same physical geographical constraints, except for their populations. Without loss of generality, we assume that country 1 is larger. The typical individual is assumed to supply one unit of labor and κ unit of capital inelastically. To rule out the Heckscher–Ohlin comparative advantage, individuals in two countries are assumed to hold the same amount of capital. Let L be the population in the space. The total amount of capital is then K = κ L. The share of labor and capital in country 1 is denoted by θ ∈ (1/2, 1). For simplification, we mainly describe the variables for country 1, whose counterparts in country 2 can be written similarly. We have a manufacturing sector under a technology of increasing returns to scale, and there is a continuum of varieties in this sector, forming the composite good M. Specifically, the composite good M1 in country 1 is ⎡ N ⎤ σ σ−1 σ −1 M1 = ⎣ d1 (i) σ di ⎦ , 0
where σ > 1 represents the elasticity of substitution between two manufactured varieties, N is the number of varieties, and d1 (i) is the demand of a typical manufactured good i in country 1. The varieties are supposed to be symmetric, so we can omit the variety name and simply use d1 to refer to the demand of each variety. The manufacturing price index in country 1 is given by ⎡ P1 = ⎣
N
1 ⎤ 1−σ
p1 (i)1−σ di ⎦
,
0
where p1 (i) is the price of a variety i in country 1. In most of the existing papers, the HME is discussed with one more agricultural sector under a technology of constant returns to scale. In contrast, in our paper, only manufactured goods are consumed in our setup. The utility function of residents in country 1 is simply M1 . Therefore, the Marshallian demand in country 1 for a variety made in country j is d j1 =
p −σ j1 P11−σ
Y1 ,
j = 1, 2,
(1)
where Y1 is the national income of country 1 and p j1 is the price of a variety made in country j and sold in country 1. Firms produce differentiated goods with identical technology.3 Specifically, following Martin and Rogers (1995), we have two production factors. Each firm needs a marginal input of labor and a fixed input of capital. We choose units of capital and 3 To keep the model as simple as possible, we do not consider asymmetric productivity, which is the main concern of the firm heterogeneity literature [see Okubo (2009) and Yu (forthcoming)].
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manufactured goods so that a fixed input of one unit of capital and a marginal input of (σ −1)/σ units of workers are required to produce one unit of a variety. The firm share in country 1 is equal to the capital share k there, so the number of firms in country 1 is k K . Assume Samuelson’s iceberg international transportation costs: τ ≥ 1 units of a manufactured good must be shipped for one unit to reach the other country. A firm located in country 1 sets prices to maximize its profit π1 = p11 d11 + p12 d12 −
σ −1 w1 (d11 + τ d12 ) − r1 . σ
(2)
We choose the labor in country 2 as the numéraire, so the wage there is w2 = 1. We then write w1 as w. The first-order condition to maximize (2) gives p11 = w and p12 = τ w. Similarly, we have p22 = 1 and p21 = τ . Therefore, the price indices in two countries are simplified as 1
P1 = {[kw 1−σ + φ(1 − k)]K } 1−σ ,
1
P2 = [(φw 1−σ k + 1 − k)K ] 1−σ ,
(3)
where φ ≡ τ 1−σ ∈ [0, 1] is the trade freeness. By the free-entry condition of firms, the operating profit earned by a typical firm is just sufficient to cover its fixed cost. We then obtain the capital rent as w 1−σ Y1 w 1−σ Y2 1 + φ , σ K w 1−σ k + φ(1 − k) φw 1−σ k + 1 − k Y1 Y2 1 φ 1−σ + . r2 = σK w k + φ(1 − k) φw 1−σ k + 1 − k r1 =
(4) (5)
Next, the national incomes are Y1 = θ K [kr1 + (1 − k)r2 ] + θ wL , Y2 = (1 − θ )K [kr1 + (1 − k)r2 ] + (1 − θ )L .
(6) (7)
In the above expressions, the first terms in the RHS are capital incomes and the second terms are wage incomes.
3 Equilibrium It is easy to conclude that we have no corner equilibrium in this model. Our analysis is, therefore, focused on the interior equilibrium, in which r1 = r2 . Equations (4) and (5) imply r1 = r2 =
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Substituting the above equalities to (6) and (7), we obtain θ L[σ w + (1 − θ )(1 − w)] (1 − θ )L[σ − θ (1 − w)] , Y2 = , σ −1 σ −1 L(θ w + 1 − θ ) r = . K (σ − 1)
Y1 =
(8)
Meanwhile, by use of r1 = r2 = r , Eqs. (4) and (5) imply (1 − φ 2 )w 1−σ Y2 = r (w 1−σ − φ)σ K , φw 1−σ k + 1 − k (1 − φ 2 )w 1−σ Y1 = r (1 − φw 1−σ )σ K . w 1−σ k + φ(1 − k) Since the left-hand sides are all positive, we obtain a lower bound and an upper bound for wage w: φ < w σ −1 <
1 . φ
(9)
Note that w = 1 is between the bounds. The above bounds imply that firms’ relocation decreases the price index in the destination country and increases that in the origin country when the trade freeness φ and wages are fixed.4 We now turn to the labor markets. Because the wage in country 1 is w, the labor demand in country 1 is equal to the total labor costs of firms there: k
σ −1 (Y1 + Y2 ) = k(θ w + 1 − θ )L , σ
where the equality is from (8). Meanwhile, the labor supply in country 1 is θ wL. The labor balance gives a simple relationship between the wage and firm share: k=
θw . θw + 1 − θ
(10)
This simple equation reveals the important fact that there is a positive correlation between two endogenous variables, w and k. A higher wage in country 1 will be associated with a higher firm share there. Meanwhile, equal wages in two countries (i.e., w = 1) happen if and only if k = θ . Many authors define the home market effect by firm share: a country with a relatively larger local demand attracts a more-than-proportionate share of IRS firms, i.e., k > θ in our notation. Meanwhile, there is an alternative definition of the HME in 4 To see this in detail, assume that one unit of firms move from country 2 to country 1. According to (3), P11−σ has an impact of w 1−σ − φ = w 1−σ (1 − φw σ −1 ) > 0, while P21−σ has an impact of φw 1−σ − 1 = w 1−σ (φ − w σ −1 ) < 0, where the inequalities are from (9). Accordingly, P1 decreases, and P2 increases.
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terms of wage that: other things being equal, the wage is higher in a larger country (Krugman 1991, p. 491; Behrens et al. 2009, footnote 1). In our notation, this can be written as w > 1. Fortunately, our simple model is able to disclose the relationship between two definitions, which has not yet been explored in the literature. For this purpose, we rewrite expression (10) as k−θ =
θ (1 − θ )(w − 1) . θw + 1 − θ
(11)
We then obtain the following conclusion from the fact of θ ∈ (1/2, 1). Proposition 1 The HMEs in terms of firm share and wage are equivalent. Without the agricultural good, the wages in two countries are not directly tied. Some existing studies investigate the wage property by models in which the HME in terms of firm share is opaque. For example, in the model of Section II in Krugman (1980), the wage in the larger country is higher but firm share is always proportional to the country size. This is because labor is the only production factor there and the labor immobility does not allow the production to respond to the demand in the manufacturing sector. In Amiti (1998), Hanson and Xiang (2004), Laussel and Paul (2007), there are several industries in the manufacturing sector. It then becomes questionable to examine the HME directly because the location of the whole manufacturing sector depends on the relative parameters of various industries. In contrast, with the help of mobile capital, our simple model clearly discloses the relationship between wage and industrial location. Intuitively, firms facing a higher wage reduce the labor shares in their total production costs,5 which exhibits the HME in terms of firm share. Generally, a higher wage in a country has two effects. On the one hand, it strengthens the market size so that more firms are likely to locate in this country to save transport costs (the market size effect). On the other hand, it has a negative effect on the production side. Firms pay the wages as production costs and a higher wage pressures more firms to exit the country (the production cost effect). The above proposition implies that the market size effect overweighs the production cost effect. To show how wage w is endogenously determined, note that the production of each firm in country 1 is d11 + τ d12 and the marginal cost in each firm is (σ − 1)/σ . The market-clearing condition of labor in country 1 is written as: θL =
σ −1 k K (d11 + τ d12 ). σ
By use of (1), (3), (8), and (10), the above equation immediately derives the following wage equation F(w) ≡ Q0 (w) + Q1 (w)φ + Q2 (w)φ 2 = 0,
(12)
5 If w > 1, the production of a firm in country 1 is then d + τ d = σ L θ + 1−θ , which is smaller 11 12 w σ −1 K σ L than the production of a firm in country 2 d22 + τ d21 = σ −1 K (1 − θ + θ w).
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w 1.8 0.6
0.8
1.6
1.4
1.2
0.2
0.4
0.6
0.8
1.0
Fig. 1 Inverted U-shape between w ∗ and φ
where Q0 (w) ≡ (w − 1)θ (1 − θ ), Q1 (w) ≡ w 2−σ [(1 − θ )w 2σ −3 − θ ]σ, Q2 (w) ≡ θ (σ − 1 + θ )w − (1 − θ )(σ − θ ). To understand the wage equation, we first focus on the case of σ = 2. Equation (12) is analytically solvable in this case, giving equilibrium wage w∗ = 1 +
2(2θ − 1)(1 − φ)φ , 2φ(1 − θ ) + θ (1 − θ ) + θ (1 + θ )φ 2
and therefore, the equilibrium firm share is k∗ = θ +
2θ (2θ − 1)(1 − θ )(1 − φ)φ θ (1 − θ )(1 − φ)2 + 2φ[1 − θ (1 − θ )(1 − φ)]
from (11). These functions are plotted in Figs. 1 and 2 for θ = 0.6 and θ = 0.8. The figures reveal some important features of the wage and firm share in country 1. First, w > 1 holds for all φ ∈ (0, 1). Despite the identical capital endowment in our model, the larger country is relatively richer than the smaller one. Secondly, both w and k have an inverted U-shape. In fact, let
φ ≡
√
2θ (1 − θ ) − θ (1 − θ ) . 2 − θ + θ2
Both w and k then increase when φ ∈ (0, φ ), and decrease when φ ∈ (φ , 1). Having complicated terms w2−σ and w 2σ −3 , Eq. (12) is not analytically solvable to obtain an explicit expression of wage w for an arbitrary σ . Nevertheless, we can show the above features analytically for any σ > 1.
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0.8
0.9
0.8
0.7
0.6
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 2 Inverted U-shape between k ∗ and φ
Lemma 1 (i) Wage equation (12) has a unique solution w ∗ , which lies in (1, ∞) for any φ ∈ (0, 1); furthermore, ∂F(w∗ )/∂w > 0 holds; (ii) Equilibrium wage w ∗ = 1 for φ = 0, 1; (iii) There is a φ ∈ (0, 1) such that both the equilibrium wage w∗ and the equilibrium firm share k ∗ increase in φ ∈ (0, φ ) and decrease in φ ∈ (φ , 1). Proof (i) is given as Lemma 6 in Appendix A, (ii) is proven in Appendix B and (iii) is given in Appendix C. The mathematical results in Lemma 1 are quite useful because they provide detailed information about both wage inequality and firm location. First, Lemma 1 (i) leads to the following conclusion: Proposition 2 For any positive and finite transport costs, the HME occurs in terms of both firm share and wage. Proposition 2 strongly supports Krugman (1980) and Helpman and Krugman (1985). Although their results are challenged by Davis (1998), our result suggests that the agricultural sector and the free trade assumption in this sector are innocuous. The HME result is consistent with Crozet and Trionfetti (2008), Zeng and Kikuchi (2009), and Takatsuka and Zeng (forthcoming), who assume an agricultural sector and positive transport costs in it. Meanwhile, the relative wage result is consistent with Krugman (1980), Amiti (1998), Hanson and Xiang (2004), and Laussel and Paul (2007), who assume multiple industries. In particular, in Corollary 1 of Amiti (1998), the author shows that the larger country is a net importer of capital for intermediate transport costs in a framework of two differentiated industries in the manufacturing sector. In our model, the HME in terms of firm share occurs, which implies that the larger country is a net exporter of manufacturing goods and an importer of capital. It is noteworthy that our result is based on the assumption of identical productivity of firms in two countries. An interesting paper of Garcia Pires (forthcoming) examines the HME with the consideration of R&D investment, which creates asymmetries between firms. As a result, he finds that a large country does not necessarily attract a more-than-proportionate share of manufacturing firms, because firms in the larger
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Table 1 Average corporate tax rates in OECD countries Years
Large (%)
Small (%)
(%)
1982
48
47
1
1983
49
47
2
1984
49
46
3
1985
48
46
2
1986
47
45
2
1987
46
43
3
1988
44
43
1
1989
43
40
3
1990
42
38
4
1991
42
36
6
1992
42
33
9
1993
41
32
9
1994
40
31
9
1995
42
31
11
1996
42
32
10
1997
42
32
10
1998
40
31
9
1999
39
31
8
2000
38
31
7
2001
36
31
5
2002
36
31
5
2003
36
30
6
2004
36
30
6
2005
35
28
7
country invest more in R&D. Nevertheless, the large country is a net exporter of the manufactured goods. The theoretical result of a net capital inflow to the larger country is consistent with the empirical study of Devereux et al. (2002) on corporate taxation. According to their updated data, we can calculate the average statutory tax rate of small OECD countries with population less than 20 million (Austria, Belgium, Finland, Greece, Ireland, Netherlands, Norway, Portugal, Sweden, and Switzerland) and large countries with population exceeding 20 million (Australia, Canada, France, Germany, Italy, Japan, Span, and the United States) in the period of 1982–2005 in Table 1. The 4th column lists the tax differential . We can clearly see that larger countries impose higher corporate tax rates, which can be seen as evidence of the agglomeration rents (Baldwin and Krugman 2004) created by the HME. Lemma 1 (i) can be used to derive more properties about the equilibrium wage w ∗ . Since w ∗ ∈ (1, ∞), both Q0 (w ∗ ) and Q2 (w ∗ ) are positive. Therefore, Q1 (w ∗ ) must be negative, giving another lower bound for w∗ as follows. w 2σ −3 <
θ . 1−θ
(13)
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While the bounds of (9) depend on φ, this bound is independent of φ. We can further clarify how the equilibrium wage w depends on the labor share θ . For this purpose, we now denote F(w) as F(w, θ ). Its partial derivative with respect to θ is ∂F(w∗ , φ) = −(2θ − 1)(w∗ − 1)(1 − φ 2 ) ∂θ −σ φ[w ∗ σ −1 − φ + w ∗ (w ∗ 1−σ − φ)] < 0, where the inequality is from (9). Therefore, according to Lemma 1 (i), we have ∂F(w ∗ , θ ) ∂θ =− > 0, ∂F(w ∗ , θ ) ∂θ ∂w
∂w∗
leading to the following conclusion. Proposition 3 The equilibrium wage in the larger country increases in its population share θ . Lemma 1 (ii) is for the cases of free trade and autarky: the wages in two countries are equal. Therefore, the HME does not occur when either φ = 0 or φ = 1. This reproduces Behrens (2005), who finds that trade is a necessary condition for the occurrence of the HME by assuming non-traded manufactured goods. However, this paper obtains the condition from the relative wage and concludes that a necessary and sufficient condition of the HME is that transport costs are positive and finite. It is interesting that the equilibrium is unique when φ = 1 in our model, which is contrastive to the infinite equilibrium result of Fujita et al. (1999) and Ottaviano and Thisse (2004). This is because the market size effect disappears when transportation is free and the production cost effect does not appear in their models assuming free transportation in the agricultural sector, which results in the same profit of a firm wherever it locates. Lemma 1 (iii) tells us how the industrial location changes with falling transport costs, which is known as the Secondary Magnification Effect (SME) in the literature. Proposition 4 The equilibrium firm share and the equilibrium wage in country 1 form inverted U-shaped curves with respect to trade freeness. Proposition 4 contributes to the literature in two respects. On the one hand, although the inverted U-pattern of spatial inequality is empirically obtained by Williamson in 1965, its theoretical supports are mostly based on industrial location (e.g., Krugman and Venables 1990, 1995; Venables 1996; Puga and Venables 1997; Zeng and Kikuchi 2009) and/or on numerical simulations (e.g., Puga 1999). The above result provides a theoretical support explicitly based on income inequality and relies on an analytical approach. This evolution pattern of location arises from the interaction of centripetal and centrifugal forces. As usual, the centripetal force comes from the increasing returns in
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the manufacturing production, which increases with the trade freeness. This force also results in the HME in terms of firm share. Meanwhile, the centrifugal force comes from the capital mobility and labor production costs. For a small φ, serving foreign markets is difficult; thus, capital and firms disperse to serve the local markets. When φ is large, serving foreign markets becomes easy; as a result, even a small wage differential will attract capital and firms moving to the cheaper country, which results in the redispersion process. The dispersion force also results in the HME in terms of wage. Evidently, under the assumption of factor-price equalization as in Helpman and Krugman (1985), the centrifugal force disappears, and the redispersion is not observed (Krugman and Venables 1990, Section 4; Fujita et al. 1999, Section 14.2). For this reason, in the literature, other assumptions which account for the unequal wages are imposed. For example, Krugman and Venables (1990, Section 5), Venables (1996), and Puga (1999) assume a decreasing-returns-to-scale technology in the agricultural sector, Krugman and Venables (1995) and Puga and Venables (1997) assume that world manufacturing demand is large enough for one country to completely specialize in manufacturing, and Zeng and Kikuchi (2009) assume differentiated agricultural goods. Here, differing from their models, we have no agricultural sector. On the one hand, the wages in two countries are not tied directly but are determined by the labor demands of the IRS market, which relies on mobile industrial firms (capital). On the other hand, the absence of the agricultural sector removes the need to consider agricultural transport costs, which is controversial among many researchers. Proposition 4 is different from the monotone property derived in Laussel and Paul (2007). Based on a one-factor model, Laussel and Paul (2007) show that the wage in country 1 declines in a monotone way with falling transport costs. Intuitively, the difference comes from the capital mobility. In the one-factor model, all rewards are distributed among immobile labor only, which results in a stronger market size effect. However, in a two-factor model, some rewards are paid to the mobile capital, which reduces the market size effect. This difference is amplified for larger transport costs. Therefore, firms disperse for large transport costs. It is noteworthy that Amiti (1998) is also based on a two-factor model, and our result is consistent with hers. It is interesting that, in the empirical data listed in Table 1, the positive roughly increases until 1995 and decreases since 1996, which supports our theoretical result of the inverted U-shape of firm share. Although we have no analytical expressions of w(φ) and w (φ) for a general φ ∈ (0, 1), we are able to provide analytical expressions for the wage derivatives w (0) and w (1). For this purpose, rewrite F(w) as F(w, φ). We then have ∂F(1, 1) ∂F(1, 0) = −(2θ − 1)σ, = (2θ − 1)σ, ∂φ ∂φ ∂F(1, 0) ∂F(1, 1) = θ (1 − θ ), = −σ (σ − 1), ∂w ∂w giving w (0) =
1 1 2θ − 1 − σ, w (1) = − . 1−θ θ σ −1
(14)
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The above shows that w (0) > 0 and w (1) < 0, which is consistent with Proposition 4. Furthermore, the facts that w (0) increases in θ and w (1) decreases in θ are consistent with Proposition 3. Moreover, the analytical expressions allow us to check the balance of payments in two countries at equilibrium. The balance of payments in country 1 is calculated as B1 = Capital and Financial Account + Trade Account = {(θ − k)K [kr1 + (1 − k)r2 ]} + [k K p12 d12 − (1 − k)K p21 d21 ] φ(w ∗ )1−σ Y2∗ = (θ − k ∗ )K r ∗ + k ∗ K [φ(w ∗ )1−σ k ∗ + 1 − k ∗ ]K φY1∗ −(1 − k ∗ )K [(w ∗ )1−σ k ∗ + φ(1 − k ∗ )]K θ (1 − θ )L F(w ∗ ). =− (σ − 1)[1 − θ + (w∗ )2−σ θ φ][θ + (w ∗ )σ −2 (1 − θ )φ] Therefore, the balance of payments is equivalent to the wage equation (12). 4 Welfare Most welfare analyses in the existing literature (e.g., Baldwin et al. 2003, Chapter 11) focus on who wins and who loses from agglomeration and whether the market produces too much or too little agglomeration. They aim to obtain optimal agglomeration. This section differs by focusing on the welfare change in the equilibrium path. The questions we try to answer are who wins and who loses in the equilibrium path and whether residents in two countries gain from trade globalization in the equilibrium path. In a traditional footloose capital model, a free-traded agricultural good is introduced, which equalizes the wages and individual incomes in two countries. Therefore, the real wage in a country is higher if and only if there are more-than-proportionate firms locating in the country. In other words, the existence of the HME implies that the larger country provides a higher real wage. In contrast, we have no agricultural sector, and the wage in country 1 varies with trade costs. On the one hand, this results in a difference in the individual incomes (wage and capital return) in the two countries. On the other hand, depending on the wage rate, the price of the domestically produced goods changes accordingly. For this reason, the higher wage in country 1 is not necessarily a merit to its residents because the locally produced goods are more expensive. Consequently, although (9) ensures that the price index is lowered by attracting more firms if wages and trade costs are fixed, the welfare deserves more detailed analysis to include both impacts from wages and trade costs. Our first question has a simple answer as follows. Proposition 5 The welfare in country 1 is higher than the welfare in country 2 for all φ ∈ [0, 1).
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Proof It is sufficient to prove that the price index is lower in country 1, which is equivalent to the following inequality according to (3): w σ −1 <
k . 1−k
(15)
Meanwhile, if σ ≥ 2, inequality (15) then holds from w σ −1 ≤ w 2σ −3 <
k θ ≤ , 1−θ 1−k
where the second inequality is (13). If σ ∈ (1, 2), inequality (15) is true again because w σ −1 ≤ w <
k θw = , 1−θ 1−k
where the equality is from (10).
Next we turn to consider the second question, which is how the welfare in two countries changes with increasing φ at the equilibrium. Note that the national incomes are given by (8) and the price indices are given by (3). The utility levels of typical individuals in two countries are calculated as 1
1
1 1 K σ −1 K σ −1 X σ −1 (w, φ), ω2 = Y σ −1 (w, φ), ω1 = σ −1 σ −1
where θ w 2−σ + φ(1 − θ ) , θw + 1 − θ θ φw 2−σ + 1 − θ Y(w, φ) = (σ − θ + θ w)]σ −1 θw + 1 − θ
X (w, φ) = [wσ + (1 − θ )(1 − w)]σ −1
are both positive. Their partial differentials with respect to φ are (1 − θ )X (w, φ) dX (w, φ) = σ −2 dφ w θ + (1 − θ )φ
(σ − 1)(σ − 1 + θ ) θ (1 − θ )X (w, φ) w (φ) − φ + w 2−σ θ + (1 − θ )φ σ w − (1 − θ )(w − 1) 1 − θ + θ w w1−σ θ [w + (w − 1)(1 − θ )(σ − 2)] + ; (16) (1 − θ + θ w)[σ w − (1 − θ )(w − 1)] dY(w, φ) θ Y(w, φ) = σ −2 dφ w (1 − θ ) + θ φ
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σ −2 w (1 − θ )[1 + θ (w − 1)(2 − σ )] θ Y(w, φ) w (φ) w σ −2 (1 − θ ) + θ φ (θ w + σ − θ )(θ w + 1 − θ ) φ (σ − 1)(σ − θ ) 1−θ . + − w θw + σ − θ θw + 1 − θ
−
(17)
The first terms of (16) and (17) are positive, indicating that all consumers benefit from cheaper imported goods by trade liberalization. The other terms summarize the relocation effects of firms, the changing price of domestically produced goods, and the varying income due to trade integration. They help us to derive the following result. Proposition 6 (i) The welfare in country 1 increases in φ when w (φ) > 0; (ii) The welfare in country 2 increases in φ when w (φ) < 0; (iii) The welfare in both countries increases in φ when φ is either small or large. Proof
(i) According to (9), we have
(σ − 1)(σ − 1 + θ ) θ − φ σ w − (1 − θ )(w − 1) 1 − θ + θ w w1−σ θ [w + (w − 1)(1 − θ )(σ − 2)] + (1 − θ + θ w)[σ w − (1 − θ )(w − 1)]
θ (σ − 1)(σ − 1 + θ ) − > σ w − (1 − θ )(w − 1) 1 − θ + θ w θ [w + (w − 1)(1 − θ )(σ − 2)] φ + (1 − θ + θ w)[σ w − (1 − θ )(w − 1)] =
(σ − 1)2 φ, 1 − θ + w(σ − 1 + θ )
which proves the monotone property of ω1 when w (φ) > 0. (ii) Similar to (i), the result is derived from Lemma 11 in Appendix D. (iii) According to the facts of w(0) = w(1) = 1 and (14), we have dX (1, 0) dφ dX (1, 1) dφ dY(1, 0) dφ dY(1, 1) dφ
= θ σ σ −1 > 0, = (1 − θ )[2θ − 1 + 2σ (1 − θ )]σ σ −2 > 0, = (1 − θ )σ σ −1 > 0, = θ [1 + 2θ (σ − 1)]σ σ −2 > 0.
Therefore, both ω1 and ω2 increase in φ at φ = 0 and φ = 1. Our results then stem from the continuity of welfare functions. The above results (i) and (ii) are understandable because the firm share in country 1 increases when w (φ) > 0 and the firm share in country 2 increases when w (φ) < 0.
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9.02
9.01
9.00
0.4
0.5
0.6
0.7
0.8
Fig. 3 A numerical counterexample
Result (iii) is more surprising. When trade costs are high, trade liberalization makes more firms move out from country 2 to country 1. This is not good for country 2 in the sense of firm share. Nevertheless, our result shows that globalization is not harmful to residents in country 2 because the gain from cheap imported goods is greater than the loss from firm relocation. Unfortunately, we cannot conclude the monotone properties of welfare for any value of φ. We have a counterexample for this. We can show that ω1 decreases in some regions of φ by a numerical simulation with parameters θ = 0.9, σ = 1.1, L = 1, and K = 1, as shown in Fig. 3.
5 Conclusion Since Krugman (1980), NTT is rapidly developed to study intra-industry trade (Arnold forthcoming) and economic integration. Different from the literature of NEG (e.g., Picard and Tabuchi 2010), in NTT, capital is an important production factor. In this paper, we simplify the well-known footloose capital model, which is widely applied to examine firm agglomeration when differentiated goods are produced under an IRS technology and transported with positive costs. The new model has only a single IRS sector without the agricultural good. The mobility of capital allows for the reaction of production to demand in the IRS sector. Such a model can derive the spatial inequalities in both industrial location and income and show the equivalence of two definitions, the HME in terms of firm share, and the HME in terms of wage. Importantly, this setup confirms the existence of the HME for all positive and finite transport costs of manufactured goods. Without the agricultural sector, the wage ratio cannot be explicitly given in our model. The results and their proofs are then less intuitive than those with the agricultural sector. Fortunately, the implicit function for the wage rate is still tractable. We are able to analytically prove the inverted U-pattern of spatial inequality: the firm share and the wage in the larger country form inverted U-shaped curves with falling transport costs.
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The above theoretical result supports the empirical result of Williamson (1965). However, it is noteworthy that there is great variance in real economic circumstances and empirical results may vary considerably if the periods of trade liberalization are not precisely determined (Ben-David 1993, 2001). In addition, as reported by Kim (2008), empirical studies about China, Mexico, and Russia show that the long-run trend of regional inequality fluctuates over time, perhaps due to poor data quality. Therefore, it is important to extend our theoretical studies in the future to reflect the real world more accurately. Finally, simple structure allows us to perform a welfare analysis and examine how large and small countries have different interests in economic globalization. We find that the welfare in the larger country increases when the industry concentrates there, and the welfare in the smaller country increases in the reindustrialization process. Furthermore, both countries gain from trade integration when transport costs are either small or high.
Appendix: A Proof of Lemma 1 (i) We prove that the solution w ∗ of F(w) = 0 is unique and lies in (1, ∞), in the following Lemmas 1–5. Lemma 6 further proves F (w ∗ ) > 0. Lemma 2 Given any φ ∈ (0, 1), equation F(w) = 0 has a solution in (1, ∞). Proof Since σ > 1, the w term of the highest order in F is either wσ −1 or w, whose coefficients are all positive. Therefore F(w) is evidently positive when w is large enough. Meanwhile, it holds that F(1) = (2θ − 1)σ φ(φ − 1) < 0. The continuity of function F(·) then ensures equation F(w) = 0 having a solution in (1, ∞). Lemma 3 Let C(w) ≡ Awa − Bw b , where A > 0, B > 0, a > b > 0, and w ∈ (0, 1]. It holds that C(w) ≤ max{C(0), C(1)}. Proof The derivative of C(w) is
C (w) = aAwa−1 − bBw b−1
⎧ 1 ⎪ bB a−b ⎪ ⎪ ⎪ , ⎨ > 0 if w > aA 1 ⎪ ⎪ bB a−b ⎪ ⎪ . ⎩ < 0 if w < aA
Therefore, sup C(w) = max{C(0), C(1)}.
Lemma 4 For θ ∈ (1/2, 1), σ ∈ (1, 3/2), and φ ∈ (0, 1) we have (2 − σ )σ φ φ 2 (σ − θ ) + θ < . 1 + (σ − 1)φ 2 − θ (1 − φ 2 ) σφ Proof Let D(θ ) = [1 + (σ − 1)φ 2 − θ (1 − φ 2 )][φ 2 (σ − θ ) + θ ].
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Its derivative is dD(θ ) = −(2θ − 1)(1 − φ 2 )2 < 0. dθ Therefore, D(θ ) > D(1) = σ φ 2 [φ 2 (σ − 1) + 1] > σ 2 φ 2 (2 − σ ).
(A.2)
The relationship between the first and the last terms of (A.2) is equivalent to (A.1). Lemma 5 Equation F(w) = 0 has no solution in (0, 1] when φ ∈ (0, 1). Proof We prove that inequality F(w) < 0 holds for all w ∈ (0, 1] and φ ∈ (0, 1). For σ ≥ 2 and w ≤ 1, we have wσ −1 ≤ 1 and w 2−σ ≥ 1. It then holds that F(w) ≤ Q1 (w)φ + Q2 (w)φ 2 ≤ (1 − 2θ )σ φ + σ (2θ − 1)φ 2 = (1 − 2θ )σ φ(1 − φ) < 0. For σ ∈ [3/2, 2) and w ≤ 1, we have wσ −1 ≤ 1 , w 2−σ ≤ 1, and w 3−2σ ≥ 1, which imply F(w) ≤ Q1 φ + Q2 φ 2 ≤ (1 − θ )σ φw σ −1 − w 2−σ [θ σ φ + (1 − θ )(σ − θ )φ 2 − θ (σ − 1 + θ )φ 2 ] = (1 − θ )σ φwσ −1 − w 2−σ [(2θ − 1)(1 − φ)σ φ + (1 − θ )σ φ] = −w2−σ (2θ − 1)(1 − φ)σ φ − (1 − θ )σ φwσ −1 (w 3−2σ − 1) < 0. Finally, we consider the case of σ ∈ (1, 3/2) and w ∈ (0, 1]. There are two subcases. (i) If σ φ < θ (1 + φ) holds, we have E ≡ (1 − θ )σ φ − (1 − θ )[φ 2 (σ − θ ) + θ ] < 0. Meanwhile, it holds that F(w) ≤ E − θ σ φw2−σ + θ [1 + (σ − 1)φ 2 − θ (1 − φ 2 )]w ≡ G(w). Since G(1) = F(1) < 0, G(0) = E < 0, we have F(w) < 0 for all w ∈ (0, 1] according to Lemma 3. (ii) If σ φ ≥ θ (1 + φ) holds, we have θ [1 + (σ − 1)φ 2 − θ (1 − φ 2 )] < θ σ φ, so that F1 (w) ≡ −w 2−σ θ σ φ + wθ [1 + (σ − 1)φ 2 − θ (1 − φ 2 )] < θ σ φ(w − w2−σ ) < 0.
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Let F2 (w) ≡ w σ −1 (1 − θ )σ φ − (1 − θ )[φ 2 (σ − θ ) + θ ], 1 2 φ (σ − θ ) + θ σ −1 w ≡ , σφ 1 σ −1 (2 − σ )σ φ w† ≡ . 2 2 1 + (σ − 1)φ − θ (1 − φ ) We then have F(w) = F1 (w) + F2 (w) and w > w † from Lemma 4. Function F1 (w) is decreasing iff w < w † according to Lemma 3, and F2 (w) > 0 iff w > w . Thus, for w ∈ (0, w ), we have F(w) = F1 (w) + F2 (w) < 0. On the other hand, for w ∈ (w , 1), both F1 (w) and F2 (w) are increasing functions, so that F(w) = F1 (w) + F2 (w) ≤ F1 (1) + F2 (1) = F(1) < 0. Therefore, F(w) < 0 holds for any w ∈ (0, 1]. Lemma 6 The root w∗ of F(w) = 0 in (1, ∞) is unique, and F (w ∗ ) > 0. Proof It is enough to prove F (w ∗ ) > 0. We have F (w) = θ (1 − θ ) + (σ − 1)(1 − θ )σ φw σ −2 + (σ − 2)θ σ φw 1−σ + θ (σ − 1 + θ )φ 2 , which is evidently positive when σ ≥ 2. For σ ∈ [2/3, 2), we rewrite F (w) = w 1−σ H(w), where H(w) = θ (1 − θ )w σ −1 + (σ − 1)(1 − θ )σ φw 2σ −3 − (2 − σ )θ σ φ +θ (σ − 1 + θ )φ 2 w σ −1 , which is an increasing function of w ∈ [1, ∞). Note that H(w) is positive when w is sufficiently large, we know that either H(w) and F (w) are positive for all w ∈ (1, ∞), or there is a w† ∈ (1, ∞) such that H(w) > 0 and F (w) > 0 iff w ∈ (w † , ∞). If F (w ∗ ) > 0 does not hold, then w ∗ ∈ [1, w † ], and F(w) decreases in [1, w ∗ ]. For this reason, F(1) > F(w∗ ) = 0, which contradicts with the known fact of F(1) < 0 (see the proof of Lemma 2). Therefore, we have F (w ∗ ) > 0 in this case. Finally, in the case of σ ∈ (1, 3/2), we have F
(w) = (σ − 1)(2 − σ )σ φw−σ (1 − θ )
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which is positive for all w ∈ [1, ∞). Since F(1) < 0 (see the proof of Lemma 2) and F(w ∗ ) = 0, we have F (w ∗ ) >
F(w ∗ ) − F(1) > 0. w∗ − 1
B Cases of autarky and free trade When φ = 1, equation F(w) = 0 degenerates to (θ w2−σ + 1 − θ )(w σ −1 − 1) = 0. Since w > 0 and 1 − θ > 0, w = 1 is evidently the unique solution to the above equation. On the other hand, when φ = 0, equation F(w) = 0 degenerates to Q0 (w) = 0, which has has only one solution w = 1 again. C Proof of Lemma 1 (iii) Let
w0 ≡ w1
=
θ 1−θ
1 2σ −3
,
w0 if σ > 3/2, ∞ if σ ∈ (1, 3/2].
Inequality (13) shows that the equilibrium wage w ∈ (1, w1 ). Lemma 7 For σ > 1, θ ∈ (1/2, 1), equation J (w) = 0 has exactly one solution in (1, w1 ), where
2θ − 1 σ w + (w − 1)(σ − θ ) 1−θ 2 θ 2 2σ −3 −σ w − . 1−θ
J (w) = 4(w − 1)w 2(σ −2) θ
Proof Simple calculation obtains dJ (w) = 2w 2σ −5 J1 (w), dw where J1 (w) ≡ 2[1 + (2σ − 3)(w − 1)]θ
2θ − 1 σ w + (w − 1)(σ − θ ) 1−θ
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σ θ −1 w+σ 2 (2σ −3) −w 2σ −3 w 1−θ 1−θ θ = 4(σ − 2)θ (σ − θ ) + (3 − 2σ )[4θ 2 − 4θ + (2 − σ )σ ]w 1−θ 4θ 2 + (σ − 1 + θ )w 2 + σ 2 (3 − 2σ )w 2σ −2 . 1−θ
+2(w − 1)θ 2
(C.1)
(C.2)
If σ ≥ 3/2, Eq. (C.1) then gives J1 (w) ≥ 0 for w ∈ (1, w1 ). If σ ∈ (1, 3/2), Eq. (C.2) then implies J1 (w) ≥ 4(σ − 2)θ (σ − θ ) +
θ (3 − 2σ )[4θ 2 − 4θ + (2 − σ )σ ]w 1−θ
4θ 2 (σ − 1 + θ )w + σ 2 (3 − 2σ ) 1−θ θ = {(σ − 1)2 [2(2θ − 1) + 2σ − 1] + 2(σ − 1)[1 − 2(1 − θ )2 ] 1−θ +(1 − 2θ )2 }w + 4(σ − 2)θ (σ − θ ) + σ 2 (3 − 2σ ) θ ≥ {(σ − 1)2 [2(2θ − 1) + 2σ − 1] + 2(σ − 1)[1 − 2(1 − θ )2 ] 1−θ +(1 − 2θ )2 } + 4(σ − 2)θ (σ − θ ) + σ 2 (3 − 2σ ) θ σ [σ (2σ − 3) + 2(2θ − 1)] + σ 2 (3 − 2σ ) = 1−θ ≥ 0. +
Therefore, J (w) increases in (1, w1 ). Furthermore, we have J (1) = −
(2θ − 1)σ 1−θ
2 < 0,
J (w0 ) = 4(w0 − 1)(w0 )2(σ −2) θ
2θ − 1 σ w0 + (w0 − 1)(σ − θ ) > 0, 1−θ
if σ > 3/2. Meanwhile, [4θ 2 (σ − 1 + θ )/(1 − θ )]w 2(σ −1) is term of w with the highest order in J (w) when σ ∈ (1, 3/2], so J (w) is positive when w is sufficiently large. Therefore, J (w) = 0 has exactly one solution in (1, w1 ). Since F(w) depends on φ, we now rewrite it as F(w, φ). Denote the solution of F(w, φ) = 0 as w(φ). We show that w(φ) takes a U-shaped form in the following Lemma 8. Lemma 8 There exists φ ∈ (0, 1) such that w(φ) increases in (0, φ ) and decreases in (φ , 1).
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Proof Since w(φ) is implicitly defined by F(w, φ) = 0, it holds that ∂F(w, φ) ∂φ w (φ) = − . ∂F(w, φ) ∂w First, w(0) = w(1) = 1 hold according to Appendix B, and we have ∂F(1, 0) = σ (1 − 2θ ) < 0, ∂φ ∂F(1, 1) = σ (2θ − 1) > 0. ∂φ Furthermore Lemma 6 shows that ∂F(w, φ)/∂w > 0 holds at equilibrium. Therefore, we know that w (0) > 0 and w (1) < 0 so that there exists a φ satisfying w (φ ) = 0. In the following, we show the uniqueness of φ so that w(φ) decreases in (0, φ ) and increases in (φ , 1). In fact, w (φ) = 0 if and only if F(w, φ) = 0, ∂F(w, φ) = 0. ∂φ
(C.3) (C.4)
Solving φ from (C.4), we obtain φ=−
Q1 . Q2
(C.5)
Note that (C.5) is positive from (13). Plugging (C.5) into (C.3), we obtain J (w) = 0, which has a unique solution according to Lemma 7. Therefore, the solution φ of (C.3) and (C.4) is unique. Lemmas for Proposition 6 Lemma 9 The equilibrium wage satisfies w 2−σ φ w−1 < . σ 1−θ
(D.1)
Proof According to the wage equation (12), we have (w − 1)θ (1 − θ ) − w2−σ θ σ φ = −(1 − θ )σ φw σ −1 − [θ (σ − 1 + θ )w − (1 − θ )(σ − θ )]φ 2 < 0. The above inequality derives (D.1).
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Lemma 10 For w ≥ 1, we have Z0 (w) > 0, where Z0 (w) ≡ θ [2θ − 1 + (1 − θ )σ ]w 2 +[3θ − 4θ 2 + (2θ 2 − 1)σ + (1 − θ )σ 2 ]w − (1 − θ )(σ − θ )(σ − 2). Proof The result follows immediately from the facts that Z0 (w) is a convex function of w and Z0 (1) = σ > 0, Z0 (1) = σ 2 (1 − θ ) + θ + (2θ − 1)σ > 0, where the last inequality is because θ ∈ (1/2, 1).
Lemma 11 Inequality Z1 (w) > 0 holds for any σ > 1, where (σ − 1)(σ − θ ) 1−θ φ w σ −2 (1 − θ )[1 + θ (w − 1)(2 − σ )] + − . Z1 (w) ≡ (θ w + σ − θ )(θ w + 1 − θ ) θw + σ − θ θ w+1−θ w Proof
(i) We first consider the case of σ > 2, in which we have (σ − 1)(σ − θ ) 1−θ σ −θ 1−θ − > − > 0. θw + σ − θ θw + 1 − θ θw + σ − θ θw + 1 − θ (D.2) Inequality Z0 (w) > 0 can be rewritten as (σ − 1)(σ − θ ) 1−θ w−1 1 + θ (w − 1)(2−σ ) + − > 0, (θ w + σ − θ )(θ w + 1−θ ) θw + σ − θ θw + 1 − θ σw
which derives the positiveness of Z1 (w) from (D.1) and the positiveness of (D.2). (ii) We now consider the case of σ ∈ (1, 2], in which the first term of Z1 (w) is positive. Therefore, Z1 (w) > 0 is true if (σ − 1)(σ − θ ) 1−θ − ≥ 0. θw + σ − θ θw + 1 − θ
(D.3)
On the other hand, if (D.3) is false, we also have Z1 (w) =
1−θ (σ − 1)(σ − θ ) w σ −1 − φ (σ − 1)2 w σ −2 + − > 0, θ w + σ −θ θw + 1 − θ θ w + σ −θ w
where the inequality is from wσ −1 ≥ 1 ≥ φ.
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