It is observed that nontradeable goods are dominant in domestic markets and international trade supplies inputs toproduce nontradeable goods. For example, if we go to an Italian restaurant in a country except Italy, we will find thatdishes which are served are nontradeable. Fresh local produce and waiter service are used as domestic inputs and Italianpasta and wine are used as imported inputs. Sanyal and Jones (1982) stated, at the beginning of their paper, “The bulkof international trade consists of the exchange of intermediate products, raw materials, and goods that require furtherlocal processing before reaching the final consumer.”
In the real world, the downstream sector which connects international trade and domestic consumers containsdistribution, construction, transport, communication, finance, food services, and business services. In Sanyal and Jones(1982), they constructed what we call a vertical production chain model to highlight such a downstream sector. Avertical production chain is a structure of production in which nonconsumable and tradeable middle products areproduced in the upstream sector and are transformed into nontradeable final consumption goods in the downstreamsector.
☆ This paper is based on my Ph.D. thesis. I would like to thank my adviser, Professor Fumio Dei, for his support and encouragement. Valuablecomments from Professor Ronald W. Jones and two anonymous on an earlier version of this paper are gratefully acknowledged. Financial assistancefrom Kobe University is appreciated.⁎ Tel.: +81 78 803 6980; fax: +81 78 803 6977.E-mail address: [email protected].
71Y. Ma / International Review of Economics and Finance 18 (2009) 70–80
In the modern economy, the downstream sector plays an important role. However, there is no study to consider howtrade theorems are altered by involving the downstream sector. In this paper, I build a two-factor model of verticalproduction chain to investigate how the Rybczynski theorem, the Stolper–Samuelson theorem, and the Heckscher–Ohlin theorem change in the setting of vertical production chain.
Many studies have followed the seminal paper by Sanyal and Jones (1982). There are two approaches to a verticalproduction chain: a trade-theoretic approach and a growth-theoretic approach. The former was adopted by Sanyal andJones (1982), Yano and Dei (2003, 2004), Takahashi (2005), and Yano, Takahashi, and Kenzaki (2006), whereas thelatter was adopted by Yano (2001, 2004), Ota (2006), and Honryo and Yano (2006).
This paper adopts the trade-theoretic approach and extend Yano and Dei (2003) by introducing capital as the secondfactor of production. A two-factor model has not been developed in the existing literature on the trade-theoreticapproach. For example, in Sanyal and Jones (1982), both the upstream sector and the downstream sector share theproperties of the specific-factor model. In Yano and Dei (2003, 2004), they assumed Ricardian technology in theupstream sector in order to focus on competition policy in the downstream sector. This paper is concerned with tradetheorems rather than competition policy.
In order to understand my vertical production chain model, it may be useful to compare my model with models inthe trade literature on nontradeable goods. In the trade literature, it is typical to introduce a nontradeable good into the2×2 HOS model, as in Komiya (1967) and Ethier (1972), for example. In this setting, tradeable goods are consumedand there is no upstream-downstream relation between sectors. By contrast, in my model, tradeable goods are middleproducts so that they cannot be consumed and a nontradeable good is produced in the downstream sector.
In my model, the total labor supply is endogenous because leisure is consumed. In the trade literature on variablelabor supply, it is typical to extend the 2×2 HOS model to incorporate variable labor supply, as in Kemp and Jones(1962), Martin and Neary (1980), and Mayer (1991), for example. In these studies, consumers consume tradeablegoods as well as leisure. In my model, however, goods that are consumed together with leisure are nontradeable.
The remainder of the paper is organized as follows. In Section 2, I construct a two-factor vertical production chain modeland analyze a small open economyunder free trade. In Section 3, theRybczynski effect and the Stolper–Samuelson effect areexamined. In Section 4, I investigate the Heckscher–Ohlin theorem. Section 5 provides some concluding remarks.
2. The model of a small open economy
The two-factor vertical production chain model is established by introducing capital as the second factor ofproduction into Yano and Dei (2003). That is, two nonconsumable middle products, A and B, are produced in theupstream sector from labor, L, and capital, K, and they are processed into one nontradeable final consumption good, F,in the downstream sector by means of labor and capital. It is assumed that the labor endowment is partly consumed asleisure and partly to be used as an input into the upstream and downstream sectors.
Let the home country be a small country. Assume that middle products A and B are freely traded in world markets.Denote by pA and pB the world prices of A and B, respectively. Let w and r be the wage rate and the rental on capital,respectively. The upstream sector is perfectly competitive and its technology is of constant returns to scale. The profitmaximization conditions in the upstream sector are expressed as
waLA þ raKA ¼ pA; ð1Þ
waLB þ raKB ¼ pB: ð2ÞThe cost-minimizing input coefficient aij≡aij (w, r) indicates the amount of input i (i=L, K) that is required to
produce one unit of the middle product j ( j=A, B).In the downstream sector, a nontradeable final consumption good F is produced by means of the middle products A
and B, labor, and capital. The downstream sector is also perfectly competitive and adopts the technology of constantreturns to scale. The profit maximization in the downstream sector implies
pF ¼ pAaAF þ pBaBF þ waLF þ raKF; ð3Þwhere pF is the price of final consumption good F. The cost-minimizing input coefficient aiF≡aiF (pA, pB, w, r)indicates the amount of input i (i=A, B, L, K) that is required to produce one unit of final consumption good F.
Fig. 1. Production curves under free trade and autarky.
72 Y. Ma / International Review of Economics and Finance 18 (2009) 70–80
The labor endowment is partly consumed as leisure and partly used in the production of the upstream anddownstream sectors. The full employment condition of labor is
aLAxA þ aLBxB þ aLFxF ¼ l; ð4Þwhere l≡L−xL denotes the economy's total labor supply, L is the labor endowment, and xL is the consumption ofleisure. xA, xB and xF represent the outputs of the middle products A and B, and the final consumption good F,respectively.
The full employment condition of capital is
aKAxA þ aKBxB þ aKFxF ¼ K; ð5Þwhere K is the capital endowment. Assume that middle product A is exported and middle product B is imported. Then,the volume of exports, E is
EuxA � aAFxF; ð6Þwhereas the volume of imports, M is
MuaBFxF � xB; ð7Þwhere aAFxF and aBFxF represent the amounts of middle products A and B that are required as inputs in the downstreamsector, respectively. The balance-of-payments condition is
pAE ¼ pBM : ð8ÞEqs. (1)–(8) show the supply side of the two-factor vertical production chain model.
From the supply side of the model, we obtain the following relationship between the output of final consumptiongood F and factor supplies1
pFxF ¼ wl þ rK: ð9ÞWe call this equation the economy-wide production function because it relates the capital endowment and theeconomy's total labor supply, that is, the sum of labor employed in the upstream and downstream sectors, to the output
1 Substituting Eqs. (6) and (7) into Eq. (8), we obtain pA(xA−aAFxF)=pB(aBFxF−xB). From this and Eq. (3), we have pAxA+pBxB=pFxF−(waLF+raKF)xF. Substituting Eqs. (1) and (2) into this, we havew(aLAxA+aLBxB)+r(aKAxA+aKBxB)=pFxF−(waLF+raKF)xF. Substituting Eqs. (4) and (5) into this yieldsEq. (9).
73Y. Ma / International Review of Economics and Finance 18 (2009) 70–80
of final consumption good. The world prices of A and B, denoted by pA and pB, are given for the small country. AsEqs. (1) and (2) show, the upstream sector is incompletely specialized. Since the upstream sector has the 2×2 HOSstructure of production, the wage rate, w, and the rental on capital, r, are fixed. From Eq. (3), the price of finalconsumption good F, pF is also fixed. Thus, the economy-wide production function is a linear function of factorsupplies. Suppose, for the present, that the capital endowment, K is constant. Under this circumstance, Eq. (9) statesthat the output of the final consumption good is a linear function of the total labor supply.
The economy-wide production curve under constant capital endowment is shown by segment HN of a straight line,T, in Fig. 1. This production curve is a straight line as long as the economy is incompletely specialized in the upstreamsector. At points H and N, complete specialization takes place.2
Production curve HN can be viewed as the transformation schedule of final consumption good F and leisure oncethe labor endowment is given. If the labor endowment is OL in Fig. 1, the consumption of leisure can be measured onthe horizontal axis to the left from point L. Since good F and leisure are nontradeable, the consumption point atequilibrium is determined by the tangency between transformation schedule HN and an indifference curve, which is notdrawn.
3. The Rybczynski effect and the Stolper–Samuelson effect
3.1. The Rybczynski effect
As is well known, the Rybczynski effect refers to the impact of changes in factor endowments on outputs ofcommodities at constant prices of factors and goods. In the 2×2 HOS model, the Rybczynski effect is considered byfocusing on the production side, because the production side can be treated independently of the consumption sideunder free trade. However, in my model, the production side cannot be treated independently of the consumption sidebecause total labor supply is endogenously determined and good F is nontradeable. In this section, we initially focus onthe consumption side to examine how xF and xL change as L and K change. Later, we will consider how xA and xBchange.
For simplicity, we assume that the representative consumer's taste is homothetic. Therefore, the ratio of thequantities consumed of good F and leisure depends only on their relative price
xFxL
¼ fpFw
� �: ð10Þ
The consumer's budget constraint is3
pFxF þ wxL ¼ wLþ rK: ð11ÞThese two equations express the demand side of the model.
Since the upstream sector is incompletely specialized, pF, w, and r are fixed under free trade. Eq. (10) implies thatxF/xL is fixed. Noting this and totally differentiating Eq. (11), we obtain
xF ¼ xL ¼ wLwLþ rK
Lþ rKwLþ rK
K; ð12Þ
where a hat (ˆ) indicates the relative change in a variable. For example, xF denotes dxF/xF. It is clear from Eq. (12) that
if LNK; then LN xF ¼ xLN K: ð13ÞChanges in endowments have an effect on the total labor supply that is endogenous in our model. From l+xL=L, we
have lL
l þ xLL
xL ¼ L. This implies
if LN xL; then l N LN xL: ð14Þ
2 This will be discussed in detail in Section 3.3 The budget constraint (11) can be derived from Eq. (9) and l≡L−xL.
74 Y. Ma / International Review of Economics and Finance 18 (2009) 70–80
From Eqs. (13) and (14), we have
if LN K; then lN LN K: ð15ÞThis shows that if the labor endowment expands more rapidly than the capital endowment, the total labor supply growsmore rapidly than both endowments.4
Such a change in the total labor supply will affect factor allocation between the upstream and downstream sectors.We denote labor and capital demands in the downstream sector by ld≡aLF xF and Kd≡aKF xF, respectively. Since aLFand aKF are fixed, we have
ld ¼ Kd ¼ xF: ð16ÞFrom the production curve Eq. (9), we have
if lN K; then lN xFN K: ð17ÞFrom Eqs. (16) and (17), we have
if lN K; then lN ld ¼ KdN K: ð18ÞThis shows that the uneven growth of factor supplies leads to the minification effect on factor demands in thedownstream sector.
Next, we turn to the factor allocation in the upstream sector. Labor and capital demands in the upstream sector aredenoted by lu≡ l−aLF xF and Ku≡K−aKF xF, respectively. From these definitions, we have (lu/l) lu+(aLF xF/l)xF= l,and (Ku/K) Ku+(aKF xF/K)xF= K. From these and Eq. (17), we obtain
if lN K; then luN l N KN Ku: ð19ÞThis shows the magnification effect of uneven growth of factor supplies on factor demands in the upstream sector.
Now, consider changes in outputs in the upstream sector, xA and xB. Assume that middle product A is relativelylabor intensive compared to middle product B. The upstream sector has the same structure of production as in the 2×2HOS model. Using the standard Rybczynski theorem, we have
if lu N Ku; then xAN luN KuN xB: ð20ÞFrom Eqs. (19) and (20), we have
if l N K; then xAN lN KN xB: ð21ÞThis shows the magnification effect of uneven growth of factor supplies on the outputs of tradeables. From Eqs. (18),(19) and (21), we can obtain the following theorem:
Theorem 1. At constant prices, the uneven growth of factor supplies has the minification effect on factor demands inthe downstream sector and the magnification effect on the factor demands in the upstream sector. Moreover, the unevengrowth of factor supplies has the magnification effect on the outputs of tradeables.
We can obtain the Rybczynski effect in our model, as follows:
xANLN xFN KN xB: ð22ÞFrom Eq. (15) and (19), we obtain that if LN K, then luN LN KN Ku. From this and Eq. (20), together with Eq. (13), weobtain Eq. (22). Eq. (22) shows that the standard Rybczynski theorem holds with respect to the outputs of the upstreamsector and a change in the output of the downstream sector is flanked by factor-endowment changes.5
4 Alternatively, if K N L, we have K N LN l . This implies that l b0 if K N0 and L =0: with the given labor endowment, capital growth bringsabout a decline in the total labor supply. This result is also found in a traditional model of variable labor supply by Mayer (1991, p. 110).5 From Eq. (22), we have xFN0 if L N K =0: an increase in one factor leads to an increase in the output of the nontradeable good. As Ethier (1972,
p. 139) points out, this result also holds in a traditional model of nontradeable goods.
75Y. Ma / International Review of Economics and Finance 18 (2009) 70–80
The total labor supply is endogenously determined. However, we can treat this as an exogenous variable if we focuson the supply side of the model. From Eqs. (17) and (21), we have
xAN lN xFN KN xB: ð23Þ
This can explain complete specialization at points H and N on the production curve in Fig. 1 where K=0. As the totallabor supply increases, the output of the labor-intensive product, xA, increases but the output of capital-intensiveproduct, xB, decreases. As a result, the upstream sector will be completely specialized in the production of the labor-intensive product at point N. Complete specialization in the capital-intensive product occurs at point H.
3.2. The Stolper–Samuelson effect
The Stolper–Samuelson effect refers to the effect of changes in the prices of commodities on factors prices. In ourmodel, the upstream sector shares the 2×2 HOS structure of production. With labor-intensive product A, we have
if pAb pB; then w b pAb pBb r: ð24ÞAs implied by Eq. (3), the relative change in the price of good F, pF, depends on ŵ, pA, pB, and r . We will show
that pF can be related only to ŵ and r . Substituting Eqs. (1) and (2) into Eq. (3) yields
pF ¼ waLF þ raKF; ð25Þwhere ãiF≡aiAaAF+aiB aBF+aiF (i=L, K) represents the sum of direct and indirect inputs of factor i required to produceone unit of good F. Totally differentiating Eq. (25), hLFwþ hKF r ¼ pF � hLFbaLF þ hKFbaKFh i
, where θLF=wãLF/pF andθKF= rãKF/pF are the sums of the direct and indirect shares of labor and capital in good F, respectively. The costminimization by the firms in the upstream and downstream sectors implieswdãLF+ rdãKF=0, i.e., hLFbaLF þ hKFbaKF ¼0 (see Appendix B for a proof). Therefore, we have
pF ¼ hLFwþ hKF r: ð26ÞThis implies
w ¼ pLb pFb r; ð27Þwhere pL denotes the price of leisure that is equal to the wage rate.
As Eq. (24) shows, changes in world prices, pA and pB, induce changes in factor prices, w and r. These factor-pricechanges alter the prices of the nontradeable goods, good F and leisure. Eq. (27) demonstrates that factor prices have thesemi-minification effect on the prices of nontradeable goods.
4. Autarky
From the Rybczynski effect shown by Eq. (22), it is not difficult for us to conjecture that a relatively labor-abundantcountry has a comparative advantage in the middle product that is labor intensive, i.e., the Heckscher–Ohlin theoremhold in our model. In this section, we will make use of the production curve under autarky to establish the Heckscher–Ohlin theorem.
4.1. The production curve under autarky
In order to derive the production curve under autarky, let us first build the two-factor vertical production chainmodel under autarky. The two-factor vertical production chain model under autarky is the same as that under free tradeexcept for E=M=0 under autarky. E=M=0 in Eqs. (6) and (7) give xA=aAFxF and xB=aBF xF. From these, Eq. (4),and Eq. (5), we have the full employment conditions under autarky as follows:
aLFxF ¼ l; ð28Þ
76 Y. Ma / International Review of Economics and Finance 18 (2009) 70–80
aKFxF ¼ K; ð29Þ
where ãLF=ãLF (w, r) and ãKF=ãKF (w, r) because ãLF and ãKF can be written as functions of w and r. Eqs. (28) and(29) imply that w/r is determined by the ratio of l/K under autarky. Thus, ãiF under autarky is affected by l/K. Note thatãiF under free trade, which is discussed in Section 3, is determined by the relative world price of middle products and isnot affected by l/K.
We derive the production curve under autarky by making use of Eqs. (28) and (29). From these two equations, xF canbe considered as a function of l and K. Suppose that the capital endowment is fixed. We can draw the production curveunder autarky as curve D in Fig. 1. This curve is concave and its slope is equal to w/pF (see the proof in Appendix A).
At point C in Fig. 1, the production curve under autarky, D, is tangent to that under free trade, T. When we movealong curve T, E and M become zero at point C. This point must lie on curve D. The slopes of these two curves aregiven by w/pF's under autarky and under free trade, respectively. Prices in the model of free trade when E=M=0 arethe same as those at point C in the model of autarky. Hence, the slopes of these curves are equal at point C. Thetangency between curve D and curve T confirms that there are gains from trade in middle products.
4.2. The Heckscher–Ohlin theorem
In order to establish the Heckscher–Ohlin theorem, we examine how the relative price of middle products pB/pA isdetermined under autarky. As under free trade, the production curve can be viewed as the transformation schedule ofgood F and leisure once the labor endowment is given. In Fig. 1, the labor endowment is OL. Point G, the point oftangency of the indifference curve I and the production curve D, shows the consumption point at equilibrium underautarky. The slope of curve D at point G determines w/pF, which determines pB/pA.
Suppose that the labor endowment increases from OL to OL′. The consumption point at the new equilibrium isdetermined as follows. An increase in the labor endowment shifts the origin of the indifference curves to the right. Afterlabor growth, point G has a smaller ratio of xF/xL than before. Due to homothetic tastes, the new indifference curvethrough point G intersects the production curve D. Point B′ is a point on line L′B′ that is parallel to line LG. This pointhas the same ratio of xF/xL as before, so that the new indifference curve through point B′ intersects curve D. The newconsumption point G′ must lie between point G and point B′ on curve D, so that w/pF must decrease.
In order to know how pB/pA changes, we consider the Stolper–Samuelson effect. Again, price Eqs. (1)–(3) holdunder autarky and thus Eq. (26) holds under autarky. We have ŵ= pLb pFb r and ŵb pAb pBb r , if product A is laborintensive. A decrease in w/pF is associated with an increase in pB/pA. It follows that labor growth leads to a rise in pB/pA if product A is labor intensive. We can conclude that a labor-abundant country has a comparative advantage in thelabor-intensive middle product.
Following Jones (1965), we can generalize this result as follows (see Appendix C for the proof):
pB � pA ¼ � hLA � hLBhKF
xLL rS þ rDð Þ
K � L� �
; ð30Þ
where θLi=waLi/pi (i=A, B). The terms σS and σD represent the elasticities of substitution between good F and leisureon the supply side and on the demand side, respectively. From (30), we can have the following Heckscher–Ohlintheorem:
Theorem 2. A relatively labor-abundant country has a comparative advantage in the middle product that is labor intensive.
5. Concluding remarks
In this paper, a two-factor vertical production chain model has been constructed. Two tradeable middle products areproduced from labor and capital in the upstream sector, and a single nontradeable final consumption good is producedin the downstream sector by means of two middle products, labor and capital. Consumers consume the finalconsumption good and leisure.
In order to simplify the analysis of trade, I have focused on a small country and assumed that the upstream sector isincompletely specialized under trade. Since the upstream sector has the 2×2HOS structure of production, all factor pricesin the small country are fixed when international prices of middle products are given. It follows that the economy-wide
77Y. Ma / International Review of Economics and Finance 18 (2009) 70–80
production curve under free trade is a straight line. This curve displays the output of the final consumption good as afunction of the total labor supply to the upstream and downstream sectors with the given total supply of capital.
Changes in factor endowments affect the outputs of tradeable products, i.e., the middle products, in the same manneras in the 2×2 HOS model. The change in output of the nontradeable final consumption good is flanked by factor-endowment changes. Since leisure is consumed, the total labor supply is affected by a change in the labor endowment. Ihave examined how factors are reallocated between the upstream and the downstream sectors when factor supplieschange. There are the magnification effect on factor demands in the upstream sector and the minification effect onfactor demands in the downstream sector.
The Heckscher–Ohlin theorem remains valid in the framework of this paper. To prove this, the production curveunder autarky has been utilized. This curve is concave and tangent to the production curve under free trade, which is astraight line. The difference in the shape of the production curve under autarky and under free trade reflects the fact thatdiminishing returns to labor occurs under autarky, but is prevented by free trade in middle products.
Appendix A
Totally differentiating Eqs. (28) and (29), we havebaLF þ xF ¼ l; ðA1ÞbaKF þ xF ¼ K: ðA2Þ
Eq. (A1) is rewritten as
xFl¼ 1�baLF
l: ðA3Þ
Subtracting Eq. (A2) from Eq. (A1), we obtainbaLF �baKF ¼ l � K: ðA4Þ
Notice that K=0when the capital endowment is fixed. SubstitutingbaLF � baKF ¼ l into the right-hand side of Eq. (A3) yields
xFl¼ 1� baLFbaLF�baKF : ðA5Þ
From hLFbaLF þ hKFbaKF ¼ 0 and
ru baKF �baLF� �= w� rð Þ; ðA6Þ
where σ is the elasticity of substitution between labor and capital for the economy-wide production, we havebaLF ¼ � hKF r w� rð Þ; ðA7Þ
and baKF ¼ hLF r w� rð Þ: ðA8Þ
We will show in Appendix B that hLFbaLF þ hKFbaKF ¼ 0 and the unit isoquant (ãLF, ãKF) has a usual shape, i.e.,0b σb∞. Substituting Eqs. (A7) and (A8) into Eq. (A5) yields
xFl¼ hLF:
78 Y. Ma / International Review of Economics and Finance 18 (2009) 70–80
Using θLF=wãLF/pF and Eq. (28), we can rewrite this as
dxFdl
¼ wpF
:
This shows that the slope of the production curve under autarky is equal to w/pF.From Eqs. (A4) and (A6), we have
r w� rð Þ ¼ K � l: ðA9Þ
This implies that if l N0 with K=0, then ŵb r . From Eq. (26), we have ŵb p Fb r . Therefore, if l N0 with K=0, thenŵ− p Fb0. This shows that the production curve under autarky is concave.
Appendix B
First, we prove hLFbaLF þ hKFbaKF ¼ 0, i.e.,
wdaLF þ rdaKF ¼ 0: ðB1ÞTotally differentiating the definition of ãiF, i=L, K, we can obtain
Substituting these equations into Eq. (B2), we obtain Eq. (B1). Notice that Eq. (B1) holds both under autarky andunder free trade, because price Eqs. (1)–(3) hold both under autarky and under free trade.
Next, in order to prove 0b σb∞, we demonstrate that the economy-wide unit isoquant is bowed-in towards theorigin under autarky. Let us first choose any different two points (aAF
t , aBFt , aLF
t , aKFt ), t=0, 1, on the unit isoquant in
the downstream sector and (aAFt , aBF
t , aLFt , aKF
t ) corresponds to ( pAt , pB
t , wt, rt ). Then we have
p0Aa0AF þ p0Ba
0BF þ w0a0LF þ r0a0KFbp
0Aa
1AF þ p0Ba
1BF þ w0a1LF þ r0a1KF: ðB3Þ
Now, we turn to the upstream sector. We assume that (aLit , aKi
t ) correspond to (wt, rt ), i=A, B, and (aLi1 , aKi
1 )≠ (aLi0 , aKi0 ).Then we have
p0A ¼ w0a0LA þ r0a0KAbw0a1LA þ r0a1KA;
p0B ¼ w0a0LB þ r0a0KBbw0a1LB þ r0a1KB:
Together with these two inequalities, the right-hand side of Eq. (B3) is rewritten as
p0Aa1AF þ p0Ba
1BF þ w0a1LF þ r0a1KFbw
0 a1LF þ r0 a1KF:
Using Eqs. (1) and (2), the left-hand side of Eq. (B3) is rewritten as
p0Aa0AF þ p0Ba
0BF þ w0a0LF þ r0a0KF ¼ w0 a0LF þ r0 a0KF:
Thus, Eq. (B3) is rewritten as
w0 a0LF þ r0 a0KFbw0 a1LF þ r0 a1KF;
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which is transformed as
w0; r0� � a 1
LF � a0LF
a1KF � a0KF
!N0:
This inequality and Eq. (B1) mean that the economy-wide unit isoquant is bowed-in towards the origin, i.e., 0b σb∞.
Appendix C
First, consider the relative supplies of good F and leisure. Substituting Eq. (A8) into Eq. (A2), we obtain xF= K−θLFσ(ŵ− r). Substituting Eq. (26) into this yields
xF ¼ hLFhKF
r pF � wð Þ þ K: ðC1Þ
Substituting Eq. (26) into Eq. (A9), we have w� pF ¼ hKFr
K � l� �
. From this and l =(LL−xLxL)/l, which is given bytotally differentiating l=L−xL, we have
xL ¼ � l
hKFxLr pF � wð Þ þ L
xLL� l
xLK: ðC2Þ
From Eqs. (C1) and (C2) we have
xF � xL ¼ r
hKFhLF þ l
xL
� �pF � wð Þ þ L
xLK � L
� �:
Defining the following as the elasticity of substitution between good F and leisure on the supply side:
rSur
hKFhLF þ l
xL
� �;
we rewrite this as
xF � xL ¼ rS pF � wð Þ þ L
xLK � L
� �: ðC3Þ
Next, consider the relative demand of good F and leisure. From Eq. (10), we obtain
xF � xL ¼ �rD pF � wð Þ; ðC4Þwhere σD represents the elasticity of substitution between good F and leisure on the demand side. From Eqs. (C3) and(C4) we obtain
pF � w ¼ � 1xLL rS þ rDð Þ�
K � L� �
: ðC5Þ
Since the upstream sector has a 2×2 HOS structure of production, we have
hLA � hLBð Þ w� rð Þ ¼ pA � pB:
In the light of this and Eq. (26), Eq. (C5) can be rewritten as Eq. (30).
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