Elastic Labor Supply, Variable Markups, and
Spatial Inequalities
著者
Hajime Takatsuka, Dao‐Zhi Zeng
journal or
publication title
Review of International Economics
volume
26
page range
1084-1100
year
2018-05-08
URL
http://hdl.handle.net/10097/00127834
Elastic Labor Supply, Variable Markups, and Spatial Inequalities
∗ Hajime Takatsuka† Dao-Zhi Zeng ‡March 13, 2018
Abstract
Assuming an inelastic labor supply, existing studies show that a larger country has a higher wage rate and a higher individual income. We reexamine these results using a model with an endogenous labor supply and variable markups. We find that these results can be reversed. Specifically, in the larger country, the wage rate is lower but the individual income is higher if the love for variety is strong and trade costs are high. In contrast, the wage rate is higher but the individual income may be lower if the love for variety is weak and trade costs are low.
Key words: Elastic labor supply, Variable markups, Wage rate, Income, Home market effect. JEL Classification: F12, F21, R12.
∗We thank two anonymous referees and Editor Hartmut Egger for valuable comments and suggestions that
have greatly improved the paper. We also thank Taiji Furusawa, Naoto Jinji, Tadashi Morita, Yasusada Murata, Takatoshi Tabuchi, Kazuhiro Yamamoto, and the participants at Asia Pacific Trade Seminars 2015, 5th Asian Seminar in Regional Science, 62nd North American Meetings of the Regional Science Association International, 29th Annual Meeting of Applied Regional Science Conference, Semiannual Meeting of Japanese Economic As-sociation 2016, and seminars at Kobe, Kyoto, Sendai, Beijing, Tianjin, Wuhan for helpful suggestions. This research was partially funded by JSPS KAKENHI Grant Numbers JP16K03629 and JP17H02514 of Japan, and the National Natural Science Foundation of China under the grant 71733001.
†Corresponding author. Graduate School of Management, Kagawa University, Saiwai-cho 2-1, Takamatsu,
Kagawa 760-8523, Japan. E-mail:[email protected]
‡Graduate School of Information Sciences, Tohoku University, Aoba 6-3-09, Aramaki, Aoba-ku, Sendai, Miyagi
1
Introduction
This paper examines the effect of an elastic labor supply on spatial inequalities across countries. Assuming an inelastic supply of labor, the existing literature suggests that the wage rate and the individual income in a larger country are both higher than those in a smaller country. We show that it is not necessarily true when the labor supply is endogenously determined.
Since the studies of Krugman (1980) and Helpman and Krugman (1985), the issue of country-size effects in intra-industry models has been examined in various settings. Many researchers have especially focused on the home market effect (HME) in terms of firm share, which is defined as a phenomenon in which a country with a relatively larger local demand attracts a more-than-proportionate share of industry with positive transportation costs (Krugman, 1980, Section III; Helpman and Krugman, 1985, Section 10.4).1 The HME is also defined in terms of wages, i.e., other things being equal, a larger country has a higher wage rate (Krugman, 1980, Section II; Krugman, 1991, p. 491). The literature shows that the HME in terms of wages is more pervasive than that in terms of firm share. For example, Davis (1998), Yu (2005), and Takatsuka and Zeng (2012b) introduce trade costs of homogeneous goods into Helpman and Krugman (1985) and show that a larger country has a higher wage rate, although it might not have a more-than-proportionate share of firms. Furthermore, this result holds even when we introduce mobile capital (Takatsuka and Zeng, 2012a; Takahashi et al., 2013) and/or preferences of the variable elasticity of substitution (VES) (e.g., Chen and Zeng, 2017; Bykadorov et al., 2015).2 In summary, it is believed that a larger country has the advantages of a higher wage
rate and a higher individual income.
In the present paper, we show that the advantages of the larger country may disappear if the labor supply is elastic and firm markups are variable. Specifically, when the love for variety is weak and trade costs are low, individual income may be lower in the larger country. Meanwhile, when the love for variety is strong and trade costs are high, a large country has a lower wage rate. We will show that these results can be attributed to the fact that the labor supply difference
across countries depends on the love for variety. The individual labor supply is larger in the
1
For example, Head and Ries (2001), Feenstra et al. (2001), and Head et al. (2002) explore the pervasiveness of the HME in terms of firm share and show that it can be reversed in cases with varieties differentiated by the nation of production.
2
Wang and Gibson (2015, Proposition 2) use a model with additively separable (AS) preferences and claim that a larger country can have a higher, a lower, or the same wage rate. However, they do not give any examples and/or conditions for the reverse HME of wages. Meanwhile, Bykadorov et al. (2015) use augmented hyperbolic absolute risk aversion (AHARA) utilities (a class of AS); through massive simulations, they show that the wage rate in a bigger country will be higher.
larger country if the love for variety is stronger. This is because variety consumption is more attractive for workers in the larger country since the country has more firms and thus they have better access to varieties. However, if the love for variety is weaker, this tendency diminishes and we may even see the opposite, i.e., the individual labor supply is smaller in the larger country. As shown later, this possibility is related to our framework based on a quasi-linear utility function generating variable markups.
In the completely free trade case, production costs (wages) are the only driver of firm location since two countries are indifferent in market access. Thus, firms eventually relocate to eliminate any wage differential. When trade costs increase marginally, the wage rate in a larger country goes up to offset its advantage in market access. However, if the love for variety is weaker, the individual labor supply in the larger country can be smaller than that in the smaller country, so that individual wage income (= wage rate× labor supply) may be lower there. In contrast, the larger country has a higher individual income when trade costs are sufficiently high. If the love for variety is strong, the individual labor supply becomes relatively large in the larger country as mentioned before, so that the wage rate (=individual income/labor supply) may be lower there.
The impacts of elastic labor supply on trade and agglomeration are explored by Ago et al. (2017, 2018) based on two different frameworks, from which our setup takes the appropriate parts to derive new findings. First, since Chen and Zeng (2017) show that variable markups are important in studying agglomeration, we borrow the quasi-linear utility function of Ago et al. (2017) to capture the procompetitive effect. Second, in their setting of one factor of labor, trade in goods is necessarily balanced so that the HME in terms of firm share is never observed. Thus, we introduce mobile capital as another production factor as in Ago et al. (2018). Since Ago et al. (2018) use a constant-elasticity-of-substitution (CES) subutility function, our results originating from variable markups are never observed in their setting. Finally, while Ago et al. (2017, 2018) focus on symmetric countries, our model is based on countries having different sizes in order to examine various home market effects.
The remainder of this paper is organized as follows. We present the model in Section 2. In Section 3, we analytically examine the country-size effects when trade costs are small. We show that the individual income in the larger country is not necessarily higher. Section 4 considers the country-size effects when trade costs are large. The result shows that the wage rate in the larger country is not necessarily higher. Finally, Section 5 is the conclusion.
2
The model
The economy consists of two countries, called countries 1 and 2, and one differentiated goods sector. The total number of workers in the world is normalized to one, and the share of country 1 is denoted as θ ∈ (1/2, 1), which implies that country 1 is the larger one. While labor is immobile between countries, capital is mobile across countries and is evenly held by all workers. The measure of the total amount of capital is assumed to be one.
Each worker is assumed to hold identical preferences, which are described by a quasi-linear utility with a quadratic subutility:
U = α ∫ n 0 x(v)dv−β 2 ∫ n 0 x(v)2dv− γ 2 [∫ n 0 x(v)dv ]2 − l, (1)
where n is the mass of varieties, x(v) is the consumption of variety v∈ [0, n], and l is the labor supply. Parameters α, β, and γ are all positive. As explained in Ottaviano et al. (2002, p.413), α represents the intensity of preferences for goods. Thus, a large α implies a large demand for the manufactured goods. Parameter β expresses consumers’ love for variety. A high β indicates that varieties are less substitutable so that the production is likely to be conducted by professional workers. We show that these two parameters play key roles in examining country-size effects. The last term of (1) implies that working generates disutility. The budget constraint is expressed as
∫ n
0
p(v)x(v)dv = wl + r≡ y,
where p(v) is the price of variety v, w is the wage rate, r is the capital return, and y is the income.
The first order condition (F.O.C.) of utility maximization gives
α− βx(v) − γ
∫ n
0
x(v′)dv′= p(v)
w , (2)
which leads to the demand function of each variety and the labor supply function:
x(v) = α β + nγ + γP (β + nγ) βw− p(v) βw (3) l = α β + nγ P w − β(S + 1)/n + γS β (β + nγ) P2 w2 − r w, (4)
where P ≡∫0np(v)dv is the price index, and S≡ 1 P2 [ n ∫ n 0 p2(v)dv− P2 ] .
If all available varieties are symmetric,3 we have p(v) = p and S = 0 so that (4) is rewritten as l = αn β + nγ p w− n β + nγ (p w )2 − r w, suggesting that ∂l ∂w = np(2p− αw) (β + nγ) w3 + r w2. (5)
As explained in Ago et al. (2017), the first term shows that a rise in w has two opposite effects on labor supply, i.e., the substitution effect and the income effect. The substitution effect is a positive effect in which a consumer increases her labor supply to reduce her leisure time and increase her consumption of varieties. Meanwhile, the income effect is a negative effect in which she reduces her labor supply: a higher wage income enables her to increase her leisure time. Equation (5) suggests that, if w/p < 2/α, the former effect dominates the latter so that the labor supply increases with w. In contrast, if w/p is sufficiently high, the labor supply decreases in w since the income effect is dominant.4 This is a backward-bending property of labor supply, which has been widely observed empirically (e.g., Nakamura et al., 1979).
The backward-bending property of labor supply is related to the variable markups, which are captured by the quasi-linear utility function employed here. Given separable utility between variety consumption and leisure, Ago et al. (2017, Appendix A) show that the backward-bending property disappears if subutility of variety consumption is described by CES, which generates constant markups, and disutility of labor is a power function of l (see Equation (3) of Ago et al. (2018)). More rigorously, the wage elasticity of labor supply is constant if the elasticity of marginal utility of variety consumption and the elasticity of marginal disutility of labor are both constant. Since the first elasticity is not constant in our quasi-linear utility function, the labor supply function is nonmonotonic in w.
In addition, the second term of (5) shows that a rise in w generates additional labor supply if workers have capital income r. Workers can save labor supply by the use of capital income. However, if w rises marginally, the individual capital income r is relatively lowered so that each worker has to supply an additional r/w2 units of labor.
For production, each firm needs a marginal input of m units of labor and a fixed input of 1 unit of capital to produce its differentiated variety. Since the total amount of capital is one, the
3
This assumption holds in our cases of autarky and free trade.
4If α is large, a rise in the wage rate tends to reduce the labor supply. This is because the constant term of
mass of firms (varieties) n is also one. Here, we impose the assumption of
α > m. (6)
As will be clear later, this ensures a positive demand of local varieties. We choose the labor in country 2 as the num´eraire; therefore, the wage rate there is one, while the wage rate in country 1 is denoted as w. As in most related papers, we assume Samuelson’s iceberg trade costs. Specifically, τ (> 1) units of the variety must be shipped for one unit to trade between two countries. Therefore, the pure profits of firms in countries 1 and 2 are
π1= (p11− mw) x11θ + (p12− mwτ)x12(1− θ) − r, (7)
π2= (p22− m) x22(1− θ) + (p21− mτ)x21θ− r, (8)
where xij and pij are the individual demand and the price in country j, respectively, for firms
located in country i. The market of the differentiated good is monopolistically competitive and each firm makes its decision of optimal price or output given the price index. From the F.O.C. of profit maximization and (3), we have the following equilibrium prices:
p11= [2β(α + m) + γm(n1+ 1)] w + τ γmn2 2(2β + γ) , p12= p22+ m 2(τ w− 1), (9) p22= [2β(α + m) + γm(n2+ 1)] + τ γmn1w 2(2β + γ) , p21= p11+ m 2(τ− w), (10)
where ni is the mass of firms in country i so that n1 + n2 = n = 1. The F.O.C. of profit
maximization also implies that
x11= p11− mw βw , x12= p12− mwτ β , x22= p22− m β , x21= p21− mτ βw . (11)
Plugging (11) into (7) and (8), zero profit conditions are written as
r = (p11− mw) 2 βw θ + (p12− mwτ)2 β (1− θ), (12) r = (p22− m) 2 β (1− θ) + (p21− mτ)2 βw θ. (13)
Meanwhile, substituting (9) and (10) into (11), we have
x11=
2βw(α− m) + γmn2(τ− w)
2βw(2β + γ) , x22=
2β(α− m) + γmn1(τ w− 1)
2β(2β + γ) .
Note that assumption (6) ensures a positive demand xii if there are no imports (nj = 0) for
i, j = 1, 2, j̸= i.
In equilibrium, the net import (resp. export) value of capital is equal to the net export (resp. import) value of varieties:
where k is the share of firms in country 1. Thus, we have n1= k and n2= 1− k.
We take w, k, and r as basic endogenous variables through which other variables, such as price and quantity, are derived. As shown in the Appendix A, variables in equations (12), (13), and (14) can be reduced to w, k, and r only. Thus, three endogenous variables, k, w, and r, are determined by these three equations, which are sufficient for us to examine how the economic system responds to the change in trade costs.
3
Country-size effects when trade costs are low
When τ = 1, the wage rate and firm share in the larger country and the capital rent are given by
wf = 1, kf = θ, rf =
(α− m)2β
(2β + γ)2 , (15)
respectively, from (12), (13), and (14). Here, the subscript “f ” indicates free trade. These sug-gest that the HMEs in terms of firm share and wages disappear. In the case of free trade, produc-tion costs (wages) are the only driver of firm locaproduc-tion, since the two countries are indifferent in market access. Thus, firms eventually relocate proportionally to country sizes so as to eliminate any wage differential. Using these results and (4), we also obtain l1= l2 = m(α− m)/(2β + γ).
The labor supply per worker is also identical across countries, and, thus, not only wage rates but also individual incomes are equalized between the two countries. In the following, we examine how these variables differ across countries if trade costs rise marginally.
3.1 Relative wage rate near free trade
By taking the total derivatives of (12), (13), and (14) with respect to τ at free trade, we obtain
dr dτ τ =1 = (α− m)βθ[α(2θ − 1) − m(3 − 2θ)] (2β + γ)2 , (16) dw dτ τ =1 = 2θ− 1, (17) dk dτ τ =1 = θ(1− θ)(2θ − 1)[ ( α2+ 3m2)β− γm (α − 3m)] m(α− m) (2β + γ) . (18)
Equation (17) shows how the relative wage rate changes when the economy is close to free trade, and (18) will be useful in Section 3.3. This implies the following result.
Lemma 1 The wage rate in the larger country is higher than that in the smaller country near
When trade costs increase, market access begins to play an important role in determining firm location. The HME literature finds that firms are expected to obtain larger profits by locating in the larger country to take the advantage of a larger market. This advantage will be offset by higher production costs there.
3.2 Labor supply difference near free trade
Next, we examine the labor supply difference ∆l ≡ l1 − l2 near free trade. Noting that the
individual labor supply (4) can be reduced to a function of k, r, w, and τ , we have
d∆l dτ τ =1 = ∂∆l ∂k dk dτ τ =1 + ∂∆l ∂r dr dτ τ =1 + ∂∆l ∂w dw dτ τ =1 + ∂∆l ∂τ τ =1 . (19)
We can show that the first term is zero by the use of (4). This suggests that at free trade, a deviation of firm location from equilibrium (15) does not impact the labor supply differential. The second term of (19) can be also shown to be zero. From (4), one unit increase in capital return saves 1/w (resp. one) unit of labor supply in the larger (resp. smaller) country. In the case of free trade, such labor saving is identical across countries, since w = 1.
According to (17), (B.3), and (B.6), the third and fourth terms of (19) are negative when
α > 2m and β is small (see Appendix B). Moreover, (17), (19), (B.3), (B.4), and (B.6) give the
following total effect of a rise in τ :
d∆l dτ τ =1 = [( α2− 2mα + 5m2)β− 2γm(α − 2m)](2θ− 1) (2β + γ)2 . Let β1 ≡ 2γm(α− 2m) α2− 2mα + 5m2.
Then, we have the following lemma:
Lemma 2 If α > 2m and β < β1, a rise in τ decreases ∆l at τ = 1; otherwise, it increases ∆l.
Intuitively, when β is small (varieties are less differentiated), workers do not care the range of consumed varieties. They rather prefer to consume cheaper goods. Near free trade, workers in the large country with a higher wage rate enjoy cheaper imported goods, so that labor supply in the small (resp. large) country is relatively increased (resp. decreased). As discussed in Appendix B, the result is closely related to the backward-bending property of labor supply. The inequality of α > 2m is a necessary condition for the backward-bending property (i.e., a negative (B.5)) to occur.
3.3 Income difference and firm location near free trade
As shown in Lemma 1, the wage rate in the larger country is higher than that in the smaller country near free trade. However, this does not necessarily imply that the individual income is higher in the larger country near free trade, since the labor supply differential depends on β as shown in Lemma 2. To be precise, we differentiate the individual income differential across countries, ∆y = wl1− l2, with respect to τ . The result at τ = 1 is
d∆y dτ τ =1 = [ ( α2+ 3m2)β− γm (α − 3m)](2θ − 1) (2β + γ)2 . (20)
Comparing (20) with (18), we find that the income differential echoes the firm share differential. This is because a country with a higher individual income attracts more firms due to its relatively large market. Their relationships are described by
dk dτ τ =1 ⋛ 0 ⇔ d∆y dτ τ =1 ⋛ 0 ⇔ β ⋛ γm(α− 3m) α2+ 3m2 ≡ β2 (< β1). (21) In summary, we have,
Lemma 3 If α > 3m and β < β2, a rise in τ decreases ∆y and k at τ = 1; otherwise, it increases them.
This lemma concludes that the smaller country has a higher individual income and a more-than-proportionate share of firms when α is large and β and τ are small. Since Krugman (1980), papers about the new trade theory declare that the larger country has an advantage in individual income, called the home market effect in terms of wages. Surprisingly, we find that as long as an elastic labor supply is incorporated, this well-known result is not necessarily true. Its occurrence depends on the intensity of preferences for manufactured goods and the love for varieties.
It is evident that ∆l < 0 is a necessary condition for ∆y < 0 to occur, since w > 1 holds. In fact, inequalities α > 2m and β < β1 in Lemma 2 are necessary conditions of α > 3m and
β < β2 in Lemma 3, since β2 < β1.
The following proposition summarizes the above results:
Proposition 1 When trade is close to free, we have w > 1. Furthermore, inequalities k > θ
and y1> y2 hold when α≤ 3m. However, when α > 3m, we have k ⋛ θ and y1 ⋛ y2 if β⋛ β2.
We know that β2 ≥ 0 for α ≥ 3m and limα→∞β2 = 0 from (21). Figure 1 plots the threshold
β2 in Proposition 1 in the α-β plane when m = γ = 1 and θ = 0.7. The dashed line is β1, which is higher than β2. In the figure, “Regular” indicates k > θ and y1 > y2, known as the
Figure 1: The near-free-trade case
The following two facts are helpful to understand the results. On the one hand, inequalities
k > θ and y1 > y2 always hold near free trade in the case of a CES subutility according to
the numerical result obtained by Ago et al. (2018) in their Section 5. This is because the backward-bending property, which plays a crucial role in Proposition 1, disappears in their constant-markup case as mentioned in Section 2. On the other hand, regardless of whether markups are constant or variable, we have k > θ and y1> y2near free trade as long as the labor
supply is inelastic (Takahashi et al., 2013; Chen and Zeng, 2017). Although a larger country has a higher wage rate for the same reason as in our case, this automatically means that the larger country has a higher income due to inelasticity of labor supply, and attracts more-than-proportionate share of firms because of its relatively larger market. Finally, mobile capital is also crucial for the inequality of k < θ. Without mobile capital, trade in goods is necessarily balanced so that trade in capital is also balanced (k = θ). Thus, our results of k < θ and y1 < y2
are the outcome of a combination of variable markups, elastic labor supply, and mobile capital.
3.4 Welfare difference near free trade
Now we examine the welfare difference. In the case of inelastic labor supply, many models in the literature show that the welfare level in the large country is higher. However, we will see that this result is not necessarily true when labor supply is elastic.
With the price expressions (9) and (10), plugging (3) and (4) into (1) obtains the indirect utility in two countries as functions of w, k, and r. They are evidently equal when trade is free. When trade is close to free, (16)–(18) give the following trend of the welfare inequality:
d(V1− V2) dτ τ =1 = (α− m)(2θ − 1)[(3m − α)β + 2mγ] (2β + γ)2 .
residents in the larger country are better off when τ is small. However, if α > 3m, then d(V1− V2) dτ τ =1 ⋛ 0 ⇔ β ⋚ 2mγ α− 3m ≡ β3 (> β1).
It is interesting that when β is high, the working hours are longer and the nominal income is also higher but the welfare is lower in the larger country.
A similar result is reported in Ago et al. (2017, Section 4.3). They find that when the marginal input in the manufacturing production is lower in a developed country than that in a developing country, then the welfare in the developed country might be lower. Capturing the variable markups is important to observe the fact because it disappears in the CES setup of Ago et al. (2018, Section 5).
4
Country-size effects when trade costs are high
This section considers the case of high trade costs. We will show that the wage rate in the larger country may be lower, although the individual income there remains higher. Furthermore, we demonstrate that the HME in terms of firm share can be reversed again for high trade costs.5
4.1 Autarky
When τ is sufficiently large, both countries cease trading all varieties, i.e., x12= x21= 0. From
(14), we immediately have k = θ, which suggests that the HME in terms of firm share disappears. In autarky, the economic activities in two countries are independent. It is not necessary for the two countries to choose the same num´eraire good. Noting that the capital rents in two countries are equal as long as trade starts, we choose the num´eraire goods in two countries to keep the equality of capital rents in the subsequent analysis of autarky.
Noting the price index Pr is nrprr in the autarky case, variety prices (9) and (10) are
pa11 = [β(α + m) + γmna1] wa 2β + na1γ = [β(α + m) + γmθ] wa 2β + θγ , (22) pa22 = β(α + m) + γmna2 2β + na2γ = β(α + m) + γm(1− θ) 2β + (1− θ)γ , (23) where the subscript “a” is reminiscent of autarky. Thus, the markup ratios in countries 1 and 2 are pa11− mwa pa11 = β(α− m) β(α + m) + γmθ, pa22− m pa22 = β(α− m) β(α + m) + γm(1− θ),
5This result is consistent with that of Chen and Zeng (2017). However, as shown in Section 3, we demonstrate
respectively. Since θ > 1/2, the markup ratio in country 1 is lower than that in country 2. This is due to the pro-competitive effect. Furthermore, the above equations show that
∂ ∂β pa11−mwa pa11 pa22−m pa22 > 0, lim β→0 pa11−mwa pa11 pa22−m pa22 = 1− θ θ < 1, βlim→∞ pa11−mwa pa11 pa22−m pa22 = 1. (24)
A larger β implies that the markup ratios in the two countries are close to each other, since each firm has a strong monopolistic power under a higher β; thus, the effects of entrants on markup ratios become weaker. Eventually, the difference of markup ratios across countries vanishes as
β → ∞.
From (11), (22), and (23), the individual consumption of a variety in both countries are
xa11 =
α− m
2β + θγ, xa22=
α− m
2β + (1− θ) γ, (25) respectively. Since xa11 < xa22, we know that the individual consumption of each variety is larger
in country 2 with fewer firms. Meanwhile, it holds that na1xa11 > na2xa22, which suggests that
the individual consumption of all available varieties is larger in country 1 with more firms. This reveals that variety consumption is more attractive for workers in country 1. Thus, the individual labor supply is larger in country 1, since the per capita consumption is equal to the per capita production. Indeed, since lj = mnjxjj (j = 1, 2) and (25), we have
la1=
m(α− m)θ
2β + θγ , la2=
m(α− m)(1 − θ)
2β + (1− θ)γ ,
which suggests that la1 > la2. A fall in β weakens this tendency because varieties are more
substitutable when β is smaller, so that the incentive to consume more in a country with more varieties becomes weaker. In fact, it holds that
∂ ∂β la1 la2 > 0, lim β→0 la1 la2 = 1, lim β→∞ la1 la2 = θ 1− θ > 1. (26)
In particular, if β → 0, the individual labor supply eventually becomes identical across countries. Finally, we examine the individual income and wage rates.6 In the case of autarky, from (7), (8), and (11), the operating profits of firms in countries 1 and 2 are expressed as
(pa11− mwa) xa11θ = pa11− mwa pa11 | {z } markup ratio × p|a11x{za11na1} expenditure = ra, (27) (pa22− m) xa22(1− θ) = pa22− m pa22 × pa22 xa22na2 = ra, (28)
6It is noteworthy that the above results hold even if capital is immobile. In contrast, the following results of
respectively. These equations show that the individual expenditure multiplied by the markup
ratio is equalized across countries through the common capital rent ra. Since the markup ratio
in country 1 is lower than that in country 2, the operating-profit equalization suggests that the individual expenditure (= individual income) in country 1 is higher than that in country 2, i.e.,
ya1> ya2.
Meanwhile, since y1 = l1w + r and y2 = l2+ r, the relative wage wa is determined by the
labor supply difference as well as the markup ratio difference across countries. Specifically, if β is sufficiently small, the markup ratio difference is significant, while the labor supply difference is negligible from (24) and (26), so that wa > 1 holds. However, if β is sufficiently large, the
reverse holds, and we have wa< 1. In other words, the regional differences in markups and labor
supply generate wage variation. In fact, from (27) and (28), we have wa=
(1− θ)(2β + θγ)2
θ [2β + (1− θ) γ]2, ra=
β(α− m)2(1− θ)
[2β + (1− θ) γ]2 . (29) We can easily show that ∂wa/∂β < 0, and
w⋛ 1 ⇔ β ⪋ γ
2 √
θ(1− θ) ≡ β4. (30)
This result generalizes some previous studies. For example, Takahashi et al. (2013) use a model with preferences of a constant elasticity of substitution (CES) and a fixed labor supply and show that w = 1 in autarky. This is because there are no differentials of markup ratios and labor supplies across countries. Meanwhile, Chen and Zeng (2017) examine the case of VES and fixed labor supply and show that w > 1 in autarky. This is because the larger country has a lower markup ratio due to the pro-competitive effect, as in our model, while the labor supply difference is not generated there. Both results can be interpreted as special cases of ours.7
In summary, we have the following lemma:
Lemma 4 In autarky, we have k = θ and y1 > y2. Furthermore, w⋚ 1 holds if β ⋛ β4.
Strict inequalities hold for the wage rates and individual incomes in autarky. Thus, even if trade in goods starts due to a fall in trade costs, these inequalities hold near autarky. Meanwhile, each country has just a proportionate share of firms in autarky. How does the firm share change if trade starts? We answer this question in the next subsection.
7Ago et al. (2018) analyzed the case of CES and an elastic labor supply to examine the possibility of symmetry
breaks. Although their main interest was in a symmetric-country case, they numerically show that a larger country has a lower wage rate in autarky. This is because each worker in the larger country supplies more labor force, as in our model, while the markup ratios are identical across countries.
4.2 Near autarky
In the CES framework, the choke prices of varieties are infinitely large so that two-way trade occurs for any finite trade costs. There is no room for one-way trade in the CES setup. In contrast, in our quasi-linear framework, the choke prices are finite, and one-way trade occurs between autarky and two-way trade.
Let τabe the minimum transport cost above which trade does not occur. If τ becomes lower
than τa, trade starts between the two countries. If the economy has only an immobile factor
(labor), both countries begin to export and import simultaneously for τ < τa because of the
trade balance. However, if the economy has a mobile factor (capital) as in our case, this does not necessarily hold. To the contrary, there is a threshold value of trade costs τow ∈ (1, τa) such
that one-way trade occurs for τ ∈ [τow, τa), while two-way trade occurs for τ ∈ [1, τow).
Let τa1≡ θ [2αβ + γm(1− θ)] [2β + (1 − θ) γ] (1− θ)m(2β + θγ)2 > 0, τa2≡ (1− θ)(2αβ + γmθ)(2β + θγ) θm [2β + (1− θ) γ]2 > 0, and g(β)≡ τa1 τa2 = ( θ 1− θ )2[ 2β + (1− θ) γ 2β + θγ ]3 2αβ + γm(1− θ) 2αβ + γmθ . Since lim β→0g(β) = ( 1− θ θ )2 < 1, lim β→∞g(β) = ( θ 1− θ )2 > 1, g′(β) > 0,
there exists a unique root of g(β) = 1, which is denoted by β5. The following lemma gives some results of β5:
Lemma 5 (i) If β < β5 (resp. β > β5), we have τa = τa2 (resp. τa = τa1) and k < θ (resp.
k > θ) for τ ∈ [τow, τa); (ii) β5 < β4 holds.
Proof. See Appendix C. ■
It is noteworthy that β ⋛ β5 if and only if τa1 ⋛ τa2. Therefore, a simple implication of
Lemma 5 is τa= max{τa1, τa2}.
According to Lemma 4, the wage rate in country 1 is higher when τ is large and β is small. Since a small β indicates consumers’ weak love for variety, the demand for varieties produced in country 2 is relatively large due to their lower prices and country 1 ceases to export its products
Regular w< 1 k< ̀ ̀4 ̀5 2 4 6 8 10 ̀ 0.1 0.2 0.3 0.4 ̀
Figure 2: The near-autarky case
to country 2 near autarky. The fact that country 2 is the net exporter of varieties suggests that it is the net importer of capital, leading to the reverse HME in terms of firm share.
From Lemmas 4 and 5, we immediately form the following proposition:
Proposition 2 Near autarky, we have y1 > y2. Furthermore, it holds that
(i) k > θ and w < 1 if β > β4; (ii) k > θ and w > 1 if β∈ (β5, β4);
(iii) k < θ and w > 1 if β < β5.
We know that β4 is independent of α from (30), while β5 decreases with α. Figure 2 shows thresholds β4 and β5 in Proposition 2 in the α-β plane when m = γ = 1 and θ = 0.7.
Finally, we derive the welfare difference. When trade is close to autarky, the welfare is always higher in the larger country. This is because
V1− V2 τ =τa = (α− m)(2θ − 1) 2(2β + γ)2[2β + γ(1− θ)](2β + θγ) {2mγ2(2β + γ)(1− θ)θ + (α − m)(3β + γ)[4β2+ 2βγ + γ2(1− θ)θ]} >0 always hold.
4.3 Summary and relevance of the results
Table 1 summarizes the results of Propositions 1 and 2. As in the figures, “Regular” in the table indicates that k > θ, w > 1, and y1> y2.
As shown in Table 1, inequalities of wage rates, individual labor supply, and incomes depend on the magnitude of β and trade costs in the present paper. Intuitively, when β is small (varieties are less differentiated), workers are not sensitive to the range of consumed varieties, and thus,
they prefer cheaper goods. If trade costs are low, the large country has a higher wage rate due to its better market access. Workers in the larger country can enjoy cheaper imported goods. This relatively increases (resp. decreases) labor supply in the small (resp. large) country. Therefore, the wage-rate (w) inequality is offset by the labor-supply (l) inequality, and, in the extreme case of free trade, individual incomes (y = w× l) are lower in the larger country. Meanwhile, when
β is large (varieties are more differentiated), the range of consumed varieties is important to
consumers. In the large country with more firms, thus, workers will increase nominal income by supplying labor force more in order to consume more varieties. If trade costs are high, however, it is difficult to expand sales in the other country, so that nominal income does not rise so much. In an extreme case of autarky, to keep the equation of y = w× l, the wage rate is lower in the larger country, and individual labor supply is negatively correlated with wage rates again.
Table 1. Country-size effects
Near free trade Near autarky
α≤ 3m α > 3m
Large β Regular Regular w < 1
Intermediate β Regular Regular Regular Small β Regular k < θ, y1 < y2 k < θ
Is it possible for the wage-rate inequality to be offset by the labor-supply inequality in the real world? Checchi et al. (2016) provides some interesting evidence on this point. They examine how income inequality is affected by inequality in working hours by use of data for the USA, the UK, Germany, and France over the period of 1989-2012. They find that wage rates and working hours move together in the USA and the UK, while they are negatively correlated and offset each other to some extent in Germany and France in 1990’s. More interestingly, income inequality in Germany expands since the end of 1990’s because wage rates and working hours turn to have positive correlation. (Checchi et al., 2016, Fig. 4).8
The empirical study of Checchi et al. (2016) corresponds to our framework of low trade costs because their paper focuses on domestic inequalities. A move from a negative correlation between wages and working hours to a positive one as in Germany/France suggests that parameter β in Germany/France has increased since the end of 1990’s. A rise in β implies that varieties are more specialized and differentiated so that the production is likely to be conducted by more
8A similar phenomenon of offsetting is observed in the US female case for the period of 1975-2002 by Gottschalk
professional workers. Thus, a plausible interpretation of their results is that professional workers are more dominant in Germany and France than the USA and the UK.
Checchi et al. (2016) are silent to where wage rates are higher and/or working hours are longer. In this respect, Rosenthal and Strange (2008) gives an interesting empirical result. When β is large (resp. small), the varieties are much differentiated (resp. substitutable) so that the production is likely to be conducted by professional (resp. nonprofessional) workers. They find a negative relationship between the working hours and the occupation-specific employment density for nonprofessionals and a positive relationship for professionals. This is consistent with our result that working hours are longer in more (resp. less) populated region when β is large (resp. small) and trade costs are low.
5
Conclusion
We reexamine the country-size effects in a model with variable markups and elastic labor supply. We show that, in contrast to existing studies, a larger country might have a lower wage rate and a lower individual income. It is clarified that variable markups and an elastic labor supply are essential to producing these results. These two factors are often neglected in typical new trade theory models (e.g., Krugman, 1980; Takahashi et al., 2013). Our contrastive results show that both of them deserve more attention in the analysis of future research on spatial inequalities across countries.
Appendix A: Reduced forms of some equations
The reduced forms of equations (12), (13), and (14) are as follows.
4β(2β + γ)2rw = θ{2αβw − m[2β + γ(1 − k)]w + mγτ(1 − k)}2 + (1− θ){2αβ − m[2βτw + γ(τw − 1)(1 − k)]}2w, 4β(2β + γ)2rw = θ{(2αβ + mγk)w − mτ(2β + γk)]2 + (1− θ){2αβ + m[γ(τw − 1)k − 2β]}2w, 4β(2β + γ)2rw(k− θ) = (1 − θ)kw[2αβ + mγ(1 − k) + m(2β + γ + γk)τw] × {2αβ − m[2βτw + γ(τw − 1)(1 − k)]} − θ(1 − k){(2αβ + mγk)w + m[2β + γ(2 − k)]τ} × [(2αβ + mγk)w − m(2β + γk)τ]
Appendix B: On the third and fourth terms of (19)
Regarding the third term of (19), we have
∂l1 ∂w τ =1 = 2pf − αwf β + γ (1− θ)m(β + γ) 2β + γ + rf, (B.1) ∂l2 ∂w τ =1 =−2pf − α β + γ θm(β + γ) 2β + γ , (B.2) ∂∆l ∂w τ =1 = m(2pf− α) 2β + γ + rf, (B.3) where pf = αβ + m(β + γ) 2β + γ (B.4)
is the common price of all varieties at τ = 1 and others are given by (15).
The first term in equation (B.1) shows the backward-bending property of the labor supply in country 1 again: the labor supply increases with w if wf/pf < 2/α, and it decreases with w
if wf/pf is large enough. More generally than (5), (B.1) includes an indirect effect in the rise
of w through price index P1. The second term of (B.1) appears because a rise in w relatively
decreases the capital rent r so that a worker has to work longer to gain the optimal consumption. Equation (B.2) shows that the opposite movement occurs in country 2: the labor supply decreases with w if 1/pf < 2/α, while it increases with w if 1/pf > 2/α. This is because a rise
in w impacts the labor supply in country 2 only through the price index. When the real wage 1/pf is low, the substitution effect is dominant, so that the rise in P2 reduces the consumption
of goods and increases leisure time. The reverse holds due to the income effect when the real wage 1/pf is high.
Reflecting the results of (B.1) and (B.2), the marginal change in the labor supply difference across countries (∆l) is expressed by (B.3). Furthermore, by use of (15) and (B.4), the equation (B.3) can be rewritten as ∂∆l ∂w τ =1 = 2m 2β− (α − 2m) mγ (2β + γ)2 + (α− m)2β (2β + γ)2 . (B.5) If α < 2m, the substitution effect is dominant since 2pf − α > 0; thus, a rise in w increases ∆l.
However, if α > 2m, the impact of w on ∆l depends on the value of β, the love for varieties. Specifically, when β is small (i.e., β < (α− 2m) mγ/[2m2+ (α− m)2]), ∆l is reduced by an increase in w. This is because the variety price p is lowered due to higher substitutability across varieties, which increases the real wage w/p so that the backward-bending property prevails.
Next, the fourth term of (19) is calculated as
∂l1 ∂τ τ =1 =−2pf − αwf β + γ (1− θ)m(β + γ) 2β + γ ,
∂l2 ∂τ τ =1 =−2pf − α β + γ θm(β + γ) 2β + γ , ∂∆l ∂τ τ =1 = m(2pf− α)(2θ − 1) 2β + γ . (B.6)
The signs in the above expressions are determined by 2pf − α, since wf = 1. Specifically,
∂l1 ∂τ τ =1 ⋚ 0 ⇔ ∂l2 ∂τ τ =1 ⋚ 0 ⇔ ∂∆l ∂τ τ =1 ⋛ 0 ⇔ 1 pf ⋚ 2 α.
Intuitively, a rise in τ raises the price indices in both countries, so that the real wage rate is relatively lowered in each country. Thus, the labor supply rises in both countries if the real wage 1/pf is high by the backward-bending property. Furthermore, in this case, the labor supply
differential ∆l decreases. The change in the labor supply is larger in the smaller country, since it imports more foreign varieties and thus impacts of a rise in τ are larger there. Specifically, from (B.4) and (B.6), if α > 2m and β is small (i.e., β < (α− 2m) γ/(2m)), then 1/pf > 2/α
holds, so a rise in τ decreases ∆l; otherwise, a rise in τ increases ∆l.
Appendix C: Proof of Lemma 5
(i) First, we assume that only country 1 exports for τ ∈ [τow, τa). When τ = τa, country
1 as well as country 2 ceases to export, i.e., x12 = 0. Thus, we have p12− mwτa = 0 from
(11). Using this equation, (9), (23), and (29), we have τa = τa1. Meanwhile, for τ ∈ [τow, τa),
consumers in country 1 have no incentive to buy any foreign goods, so that the F.O.C. (2) of utility maximization does not hold. Instead, we have the condition of a choke price, that
α− γn1x11 < p21/w at τ = τa = τa1. Using (22), (25), (29), and the fact that n1 = θ at
autarky, we can show that this inequality is equivalent to τa1 > τa2 at τ = τa = τa1. In
summary, if country 1 exports for τ ∈ [τow, τa), we have τa = τa1 > τa2. In a similar way, we
can show that if country 2 exports for τ ∈ [τow, τa), we have τa= τa2 > τa1.
By the contraposition of these two facts, country 2 (resp. country 1) exports for τ ∈ [τow, τa)
and τa= τa2 (resp. τa= τa1) if and only if τa2 > τa1 (resp. τa1 > τa2). Furthermore, from the
property of g(β), it holds that τa2 > τa1 (resp. τa1 > τa2) if and only if β < β5 (resp. β > β5).
Finally, from the trade and capital balance, the fact of one-way trade from country 2 to country 1 suggests that country 2 imports capital so that the HME in terms of firm share is reversed; namely, k < θ.
(ii) Since function g(β) depends on α, we know that β5 also depends on α. We use notations
g(β, α) and β5(α) to emphasize these facts. It is readily verified that g(β, α) increases in both
β5(m) = β4 due to g(β4, m) = 1. Therefore, we have the following relationships for a general α
under (6): g(β5(α), α) < g(β5(m), α) = g(β4, α), which give β5 < β4. ■
References
Ago, T., Morita, T., Tabuchi, T., and Yamamoto, K. (2017) Endogenous labor supply and international trade. International Journal of Economic Theory 13, 73-94.
Ago, T., Morita, T., Tabuchi, T., and Yamamoto, K. (2018) Elastic labor supply and agglom-eration. Journal of Regional Science 58, 350-362.
Bykadorov, I., Molchanov, P., and Kokovin, S. (2015) Elusive pro-competitive Effects and harm from gradual trade liberalization. mimeo.
Checchi, D., Garc´ıa-Pe˜nalosa, C., and Vivian, L. (2016) Are changes in the dispersion of hours worked a cause of increased earnings inequality?, IZA Journal of European Labor Studies 5:15, 1-34.
Chen, C.-M. and Zeng, D.-Z. (2017) Home market effects: beyond the constant elasticity of substitution. Journal of Economic Geography, forthcoming.
Davis, D. R. (1998) The home market effect, trade and industrial structure. American Economic
Review 88, 1264-1276
Feenstra, R.C., Markusen, J.R., and Rose, A.K. (2001) Using gravity equation to differentiate among alternative theories of trade. Canadian Journal of Economics 34, 430-447.
Gottschalk, P. and Danziger, S. (2005) Inequality of wage rates, earnings and family income in the United States, 1975–2002. Review of Income and Wealth 51, 231–254.
Head, K., Mayer, T., and Ries, J. (2002) On the pervasiveness of home market effects. Economica 69, 371-390.
Head, K. and Ries, J. (2001) Increasing returns versus national product differentiation as an explanation for the pattern of US-Canada trade. American Economic Review 91, 858-876.
Helpman, E. and Krugman, P. (1985) Market Structure and Foreign Trade, Cambridge, MA: MIT Press.
Krugman, P. (1980) Scale economies, product differentiation, and the pattern of trade. American
Krugman, P. (1991) Increasing returns and economic geography. Journal of Political Economy 99, 483-499.
Nakamura, M., Nakamura, A., and Cullen, D. (1979) Job opportunities, the offered wage, and the labor supply of married women. American Economic Review 69, 787-805.
Ottaviano, G., Tabuchi, T., and Thisse, J.-F. (2002) Agglomeration and trade revisited.
Inter-national Economic Review 43, 409-436
Rosenthal, S.S. and Strange, W.C. (2008) Agglomeration and hours worked. The Review of
Economics and Statistics 90, 105-118.
Takahashi, T., Takatsuka, H., and Zeng, D.-Z. (2013) Spatial inequality, globalization and foot-loose capital. Economic Theory 53, 213-238.
Takatsuka, H. and Zeng, D.-Z. (2012a) Mobile capital and the home market effect. Canadian
Journal of Economics 45, 1062-1082.
Takatsuka, H. and Zeng, D.-Z. (2012b) Trade liberalization and welfare: Differentiated-good versus homogeneous-good markets. Journal of the Japanese and International Economies 26, 308-325.
Wang, X. and Gibson, M.J. (2015) Trade, non-homothetic preferences, and the impact of country size on wages. Economics Letters 132, 121-124.
Yu, Z. (2005) Trade, market size, and industrial structure: revisiting the home market effect.
