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A model that derives conditions under which theorems regarding output effects of factor endowments are valid, focusing on the differences from the neoclassical framework. It explains how governments can correct distortionary effects of externalities. the production technology, marginal products of factors, and production stability.
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(^1) Department of Economics, Box 353330, University of Washington, Seattle, WA 98195-
3330, U. S. A.; Tel. 1-206-685-1859; Fax: 1-206-685-7477; e-mail: karyiu@u.washington.edu; http://faculty.washington.edu/karyiu/
Abstract
This paper examines the validity of the ¯ve fundamental theorems of international trade and some other issues in a general model of externality. The model allows the possibility of own-sector externality and cross-sector externality. This paper derives conditions under which some of these theorems are valid, and explains what the government may choose to correct the distortionary e®ects of externality. Conditions under which a economy with no optimality policies may gain from trade are also derived.
°c Kar-yiu Wong
can be obtained. As is done in Wong (2000b), we will examine the validity of the ¯ve fundamental theorems of international trade, but we are also using the present model to derive more implications of externality. Noting that externality is a form of distortion, meaning that a competitive equilibrium in the absence of any corrective policies is in general suboptimal, we do two more things in the present paper: deriving the optimal tax-cum-subsidy policies to correct the distortion, and analyzing the gains from trade when no corrective policies have been imposed. The results we will derive are more general than those in the literature. Section 2 of the present paper introduces the model and its basic features. The virtual system technique in analyzing comparative static results is also introduced. Section 3 derives the output e®ects of an increase in factor endowments. The validity of the Rybczynski Theorem will be discussed. Section 4 turns to the relationship between commodity prices and factor prices, with a careful examination of the validity of the Stolper-Samuelson Theorem. Autarkic equilibrium is examined in Section 5, where the optimal policy will be derived. Section 6 allows international trade between two countries with identical technology and preferences, but the countries may have di®erent factor endowments. The validity of the Law of Comparative Advantage, the Heckscher-Ohlin Theorem and the Factor Price Equalization Theorem will be examined. Section 7 analyzes the gains from trade for one or both economies. Su±cient conditions for a gainful trade will be derived. The last section concludes.
2 The Model
The model examined in the present paper is a general version of the basic model of externality introduced in Wong (2000b, 2000c). The present model is used to examine how some of the results derived in Wong (2000b) may or may not survive in a more general model. In particular, we will examine whether the ¯ve fundamental trade theorems are still valid. The model will also be used to derive more properties of externality that have not been examined in Wong (2000b). In this section, we brie°y describe the model and explain how it is di®erent from the basic model.
There are two countries, but for the time being, we focus on the home economy. There are two goods, labeled 1 and 2, and two factors, capital and labor. The technology
of sector i; i = 1; 2, is described by the following production function:
Qi = hi(Q 1 ; Q 2 )Fi(Ki; Li); (1)
where Qi is the output of sector i; and Ki and Li are the capital and labor inputs employed in the sector. Function hi(Q 1 ; Q 2 ) measures the externality e®ect. Firms in sector i regard the value of the function as constant, but in fact it depends on the aggregate output levels of the sectors. The argument Qi in function hi(:; :) measures the own-sector externality e®ect while Qk; i 6 = k; represents the cross-sector exter- nality e®ect. Function Fi(:; :) is increasing, di®erentiable, linearly homogeneous, and concave. We assume that sector 1 is capital intensive at all possible factor prices. Denote the supply price of sector i by psi : Let us de¯ne the following elasticities of function hi(:; :)
"ij ´
@hi @Qj
Qj hi
= hij
Qj hi
; i; j = 1; 2 ;
where hij ´ @hi=@Qj : The own-sector elasticity of sector i; "ii; is sometimes called the sector's rate of variable returns to scale. The sign of hij ; or that of "ij ; represents how a change in the output of good j may a®ect hi(:; :): For example, if "ii > 0 ; sector i is subject to increasing returns; if "ij > 0 ; i 6 = j; then an increase in the output of good j will have a positive externality on the production of good i: The current model reduces to some special cases. For example, if " 12 = " 21 = " 22 = 0 for all output levels, it reduces to the basic model in which only externality in sector 1 exists with no cross-sector externality. The rest of the model has the usual properties of a neoclassical framework; for example, all sectors are competitive and there are perfect price °exibility and perfect factor mobility across sectors. In the present framework, full employment of factors is achieved
K = K 1 + K 2 (2) L = L 1 + L 2 ; (3)
where K and L are the available amounts of capital and labor in the economy, and are assumed to be given exogenously.
Since the ¯rms in the two sectors take the value of hi = hi(Q 1 ; Q 2 ) as given, the private marginal product (PMP) of factor j in sector i; i = 1; 2, j = K; L; is equal to
P M Pij = hiFij (Ki; Li); (4)
Condition A. 1 ¡ "ii > 0 ¸ "ik; and (1 ¡ " 11 )(1 ¡ " 22 ) > " 12 " 21 ; i 6 = k:
Lemma 1. Given condition A, the social marginal product of labor or capital in any sector is positive.
The lemma can be proved by using (8) and (9). Condition A is a su±cient condition for positive social marginal products. It appears to be restrictive, but generally what it requires is that the externality terms are not too signi¯cant in magnitudes. Note that the relationship between the e±cient outputs in the present model can be described by the production possibility frontier, which is negatively sloped, but its curvature is in general ambiguous.
Following Wong (2000b), we introduce the concept of virtual and real systems. In the virtual system, we de¯ne ~Qi as the virtual output, which is equal to ~Qi = Fi(Ki; Li) with the later function described in (1). Since function Fi(:; :) has the properties of a neoclassical production function, the virtual system behaves like a neoclassical framework. The advantage of the present approach is that the properties of the neoclassical framework are well known. Let us denote the price of sector i in the virtual system by ~pi. As it is done for the neoclassical framework, the virtual output of sector i can be expressed as a function of virtual prices and factor endowments:^3 Q~i = ~Qi(~p 1 ; p~ 2 ; K; L). The virtual output of sector i is related to the real output in the following way:
Qi = hi(Q 1 ; Q 2 ) ~Qi(~p 1 ; ~p 2 ; K; L): (11)
Similarly, the virtual prices are related to the real prices in the following way:
p~si = hi(Q 1 ; Q 2 )psi : (12)
Conditions (11) and (12) give the relationship between the virtual and real systems. The approach we take in this paper is to make use of the properties of the neoclassical framework and then apply conditions (11) and (12) to derive the properties of the real system.
(^3) We can de¯ne a virtual GDP (gross domestic product) function. Partial di®erentiation of the function with respect to virtual prices gives the virtual output function. See Wong (1995) for more details.
The e±cient virtual outputs can be described by a virtual production possibility frontier, which is negatively sloped and concave to the origin. For the analysis below, the frontier is assumed to be strictly concave to the origin. The output function in (11) can be used to de¯ne the following elasticities:
´ik =
@ Q~i @ p~k
p ~k Q^ ~i
3 Output E®ects
In this section, we examine the validity of the two fundamental trade theorems in the present framework: the Rybczynski Theorem and the Stolper-Samuelson Theorem.
Di®erentiate conditions (11) and (12) totally and rearrange the terms to give
dQi = hii Q~idQi + hij Q~idQj + hi[ ~Qiid~psi + ~Qij d~psj + ~QiK dK + ~QiLdL] (13) d~psi = hiipsi dQi + hij psi dQj + hidpsi ; (14)
where i 6 = j: Combining together equations (13) and (14), we have
'iidQi + 'ij dQj = h^2 i Q~iidpsi + hi Q~ij hj dpsj + hi Q~iK dK + hi Q~iLdL; (15)
where
'ii = 1 ¡ hii Q~i ¡ hi Q~iihiipsi ¡ hi Q~ij hjipsj
'ij = ¡(hij Q~i + hi Q~iihij psi + hi Q~ij hjj psj ):
Let us introduce the following elasticities:
´ij =
@ Q~i @ p~sj
p ~sj Q^ ~i
hihj Q~ij psj Q^ ~i
´im =
@ Q~i @m
m Q^ ~i^ =^
him Q~im Q^ ~i^ ;
where i; j 2 f 1 ; 2 g; and m 2 fK; Lg: From the properties of the neoclassical frame- work, we know that Q~ij > 0 for i = j; or < 0 for i 6 = j: This implies that ´ij > 0
Denote the determinant of the matrix in (23) by © ´ ® 11 ® 22 ¡ ® 12 ® 21 : Assuming that © 6 = 0; condition (23) can be solved for the changes in output levels:
Q^ 1 = [(´ 11 ® 22 ¡ ´ 21 ´ 12 )^ps 1 + (´ 12 ® 22 ¡ ´ 22 ´ 12 )^ps 2
Q^ 2 = [(´ 21 ® 11 ¡ ´ 11 ® 21 )^ps 1 + (´ 22 ® 11 ¡ ´ 12 ® 21 )^ps 2
De¯ne z ´ Q 1 =Q 2 as the output ratio, and ps^ ´ ps 1 =ps 2 as the supply price ratio, where good 2 is chosen as the numeraire. Conditions (24) and (25) can be combined to give
z^ =
p^s^ +
where
¹ = ´ 11 (® 21 + ® 22 ) ¡ ´ 21 (® 11 + ® 12 ) (27) ¾ = ´ 1 K (® 21 + ® 22 ) ¡ ´ 2 K (® 11 + ® 12 ) (28) ³ = ´ 1 L(® 21 + ® 22 ) ¡ ´ 2 L(® 11 + ® 12 ): (29)
It should be noted that if all ®ij > 0 ; i; j = 1; 2 ; then ¹; ¾ > 0 and ³ < 0. However, because the signs of ®ii and ®ij are ambiguous, ¹; ¾; and ³ may be positive or negative. We note further that
¾ + ³ = (® 21 + ® 22 ) ¡ (® 11 + ® 12 ): (30)
In (30), the sign of ¾ + ³ is ambiguous, even if all ®ij > 0 : Equation (26) shows the dependence of the output ratio on prices and factor endowments. Speci¯cally, we say that the price-output response is normal if and only if ¹=© > 0 ; and the capital-output (labor-output) response is normal if and only if ¾=© > 0 (³=© < 0): Note that an alternative form of (26) is
z^ =
p^s^ +
In (31), (¾ + ³)=© can be termed the scale e®ect of an increase in factor endowment. To see this point, suppose that the economy expands uniformly so that ^K = ^L > 0 ; and that the supply price is ¯xed. Then (31) reduces to:
z^ =
Condition (32) represents the scale e®ect of an increase in factor endowments on the output ratio. Conditions (27) to (30) consist of the following two terms: (® 11 + ® 12 ) and (® 21 + ® 22 ): Let us for the time being focus on these two terms. Making use of (21) and (22), we have
® 21 + ® 22 = 1 ¡ (" 21 + " 22 ) ¡ ´ 22 (" 22 + " 21 ) ¡ ´ 21 (" 11 + " 12 ) = 1 ¡ (" 21 + " 22 ) + ´ 22 (" 11 + " 12 ¡ " 21 ¡ " 22 ): (33)
We can obtain a similar expression:
® 11 + ® 12 = 1 ¡ (" 11 + " 12 ) ¡ ´ 11 (" 11 + " 12 ¡ " 21 ¡ " 22 ): (34)
Making use of (16), (17), (33), and (34), equation (27) reduces to
¹ = ´ 11
s 1 s 2
s 2
[1 ¡ s 1 (" 11 + " 12 ) ¡ s 2 (" 21 + " 22 )] : (35)
Similarly, equation (28) and (29) reduce to
¾ = ¡
j¸j
¡ ´ 11 (s 2 ¸L 1 ¡ s 1 ¸L 2 )(" 11 + " 12 ¡ " 21 ¡ " 22 )=s 2 ] (36)
³ =
j¸j
¡ ´ 11 (s 2 ¸K 1 ¡ s 1 ¸K 2 )(" 11 + " 12 ¡ " 21 ¡ " 22 )=s 2 ]: (37)
Summing up conditions (36) and (37), we have
¾ + ³ = (" 11 + " 12 ¡ " 21 ¡ " 22 )(1 + ´ 11 =s 2 ): (38)
Condition (38) means that the scale e®ect is zero if " 11 + " 12 = " 21 + " 22 : Consider the following condition:
Condition B. " 11 + " 12 > " 21 + " 22 :
Lemma 2. Given condition A, ¹ > 0 : Given conditions A and B, ¾; ¾ + ³ > 0 and ³ < 0 :
This lemma can be proved easily by making use of equations (35) to (38). Note that condition A is su±cient for ¹ to be positive.
corresponding supply price ratio is equal to p^0 ; as depicted in the diagram, then ac- cording to the adjustment rule in (39), z will drop until point A is reached. This is the new equilibrium, which is stable. In fact, (26) and (40) show that there is a correspondence between local stability and price-output response: an equilibrium is locally stable if and only if the price-output response is normal. This correspondence can also be illustrated in Figure 1.^7 In the basic model analyzed in Wong (2000b), stability of an equilibrium requires that © be positive. However, in the present model, a positive © is no longer a necessary or su±cient condition.
We now make use of the adjustment rule introduced to analyze factor-output re- sponses. Let us focus on the e®ects of an increase in the capital endowment, as the e®ects of an increase in labor endowments can be analyzed in the same way. If we keep z constant, the relationship between ps^ and K is given by a shift of the supply price schedule as a result of an increase in the capital endowment. In particular, the relationship is given by the following condition
p^s^ = ¡
Four cases, depending on how an increase in K may a®ect the price level, and how the price level a®ects the output, are shown in panels (a) to (d) of Figure 2. For example, panels (a) and (b) shows the cases of normal price-output response, while panels (c) and (d) illustrate perverse price-output response. Furthermore, panels (a) and (c) of Figure 2 represent the cases in which the supply price schedule shifts down due to an increase in K: This happens when ¾=¹ > 0 : Similarly, if ¾=¹ < 0 ; an increase in K will shift the supply price schedule up, as shown in panels (b) and (d). Since ¾; ¹; and © can be positive or negative, how output may respond to changes in the supply price and the capital endowment is not clear. In the following table, we consider all possibilities. Table 1 shows the types of price-output response, capital-output response, local and global stability, depending on the signs of the three variables, ©; ¾; and ¹: Column (5) shows the price-output response, which may be normal (if ¹=© > 0) or perverse (if ¹=© < 0). We showed earlier that the production equilibrium is locally stable if the price-output response is normal. Column (6) gives the capital-price response. We
(^7) This is what Samuelson (1949) called the Correspondence Principle. For more details, see Wong
(2000b). See Samuelson (1971) for the Global Correspondence Principle.
© ¹ ¾ p-o k-p k-o global panel 1 + + + n n n n (a) 2 + + ¡ n p p p (b) 3 + ¡ + p p n p (d) 4 + ¡ ¡ p n p n (c) 5 ¡ + + p n p n (c) 6 ¡ + ¡ p p n p (d) 7 ¡ ¡ + n p p p (b) 8 ¡ ¡ ¡ n n n n (a)
n = normal response; p = perverse response
Table 1: Output Responses to Prices and Capital Endowments
say that it is normal if an increase in the capital endowment will cause a drop in the supply price at any given output ratio, i.e., if ¾=¹ > 0; otherwise, it is perverse if ¾=¹ < 0 : The local capital-output responses are given in column (7). Because sector 1 is capital intensive, we say that the capital-output is normal if an increase in the capital endowment induces locally an increase in z: A normal response is achieved if ¾=© > 0 ; or the response is perverse if ¾=© < 0 : Column (8) shows the results if ¯nite changes are allowed. The last column refers to the panels of Figure 2. Columns (7) and (8) are what our current focus is. A normal local capital-output response indicates that the Rybcyznski e®ect holds.^8 As shown in Table 1, the Ryb- cyznski e®ect is not guaranteed. In the present basic model, it is argued that if ¯nite changes are allowed, the Rybcyznski e®ect will hold globally even if it is not locally. We now examine whether such results are still true. Column (8) of Table 1 shows that they are no longer true: In fact, they are true in four of the eight cases only: cases 1, 4, 5, and 8. Alternatively, we say that they are true only in panels (a) and (c) of Figure 2. Let us examine these cases more carefully. Suppose that the exogenously given price ratio is p^1 ; with the initial equilibrium at point A or B, at which the price line at p^1 cuts the initial (solid) supply price schedule. In panel (a) of Figure 2, which describes cases 1 and 8 in Table 1, the supply schedule is positively sloped, and an increase in the capital stock shifts the schedule down, lowering the supply price at
(^8) This means that a small increase in capital will induce a local increase in the output of good 1
relative to good 2.
The e®ects of an increase in the size of the economy on the output and the autarkic price ratio can be derived from (31) and (32). For example, (32) shows the e®ect of an increase in the size of the economy on the output under a given supply price ratio. The response in a local sense is normal if (¾ + ³)=© > 0 : A comparison of this case with the case of an increase in the capital endowment shows that the previous analysis can be applied here. For example, a table similar to Table 1 can be constructed, with column 4 representing ¾ +³: Whether a local output response is normal, and whether a global change is normal, can also be derived in the same way. We thus have
Proposition 3 A normal local size-output response is neither necessary nor su±cient for a normal global size-output response. A size-output response is normal locally if (¾ + ³)=© > 0 : A size-output response is normal globally if the size-price is normal, irrespective to the sign of ©: Given conditions A and B, the size-output response is always normal in a global sense.
Note that, as explained earlier, if " 11 + " 12 = " 21 + " 22 ; there is no scale e®ect so that a uniform increase in the size of the economy will not a®ect the autarkic price ratio (with homothetic preferences).
4 Factor Price E®ects
Keeping good 2 as the numeraire, we now examine how a change in the supply price ratio ps^ may a®ect factor prices. In this section, we assume that factor endowments are ¯xed. Suppose that there is an increase in the supply price of good 1 while the supply price of good 2 (the numeraire) is ¯xed, i.e., ^ps 1 > 0 while ^ps 2 = 0. Rearrange the terms in (14) to give
bp~s 1 = " 11 Q^ 1 + " 12 Q^ 2 + ^ps 1 (42) bp~s 2 = " 21 Q^ 1 + " 22 Q^ 2 : (43)
Equations (42) and (43) show that even though the relative price of good 2 is ¯xed, a change in the relative price of good 1 could change the virtual prices of both goods due to a change in the outputs of the sectors. Substitute the changes in outputs given in (24) and (25), setting the changes in ps 2 ; K; and L to zero, into the above
two equations to give
bp~s 1 = ± 1 p^s 1 (44) bp~s 2 = ± 2 p^s 1 ; (45)
where
± 1 = [" 11 (´ 11 ® 22 ¡ ´ 21 ® 12 ) + " 12 (´ 21 ® 11 ¡ ´ 11 ® 12 ) + 1] ± 2 = [" 21 (´ 11 ® 22 ¡ ´ 21 ® 12 ) + " 22 (´ 21 ® 11 ¡ ´ 11 ® 12 )] :
Equations (44) and (45) show explicitly how a change in the real prices of the goods could a®ect their virtual prices. This result has the following implications on the corresponding changes in factor prices.
Lemma 3. If own-sector externalities are non-negative while cross-sector external- ities are non-positive ("ii ¸ 0 and "ij · 0 ; i; j = 1; 2; i 6 = j); and if ®ij ¸ 0 for i; j = 1; 2 ; with at least one inequality, then ± 1 ¸ 1 ; ± 2 · 0.
Lemma 3 follows immediately from the de¯nitions of ± 1 and ± 2 : Note that if ®ij = 0 for all i; j = 1; 2 ; then ± 1 = 1 and ± 2 = 0: From (44) and (45), Lemma 3 implies that given the conditions stated an increase in the relative price of good 1 will raise the virtual relative price of good 1 but lower (or not raise) the virtual relative price of good 2. De¯ne the unit cost function of sector i in the virtual system by ci(w; r); i = 1; 2 ; which is linearly homogeneous, di®erentiable, and concave. With positive outputs of both goods, the cost-minimization conditions are
ci(w; r) = ~psi : (46)
Di®erentiate both sides of (46), rearrange terms, and make use of (42) and (43) to yield
"
w^ ^r
p ^s 1 ; (47)
where #ij is the elasticity of unit cost function of sector i with respect to the price of factor j: The determinant of the matrix in (47) is equal to D = # 1 L# 2 K ¡ # 2 L# 1 K < 0 ;
where μ ´ 1 =(±©+¹): Condition (51) shows how the autarkic price ratio is dependent on the factor endowments. It is clear from the condition that the factor endowment e®ects depend on the magnitudes and signs of variables ¾; ³; and μ: For example, suppose that the economy has an increase in its capital stock. The resulting change in the autarkic price ratio is
p^a^ = ¡μ¾ K:^ (52)
Thus, a small increase in the capital stock will lower the autarkic price ratio if sign(μ) = sign(¾): We say that in this case the response of the autarkic price ratio is normal. On the other hand, if ¾ > 0 under the conditions mentioned in Lemma 2, then, as argued in Wong (2000b), an increase in the capital stock will lower in a global sense the autarkic price ratio, whether or not μ is positive. Similar conclusion can be reached for an increase in the labor endowment, or a uniform increase in the size of the economy. Thus we have
Proposition 5 A locally normal response of the autarkic price ratio to an increase in the endowment of either capital or labor, or to a uniform increase in the size of the economy, is neither necessary nor su±cient for a globally normal response of the autarkic price ratio to an increase in the capital stock. Given conditions A and B, the response of the autarkic price ratio to an increase in the factor endowment is globally normal.
Externality in the present model represents a distortion. This means that a market equilibrium in general is not optimal in terms of the welfare of the economy. From the description of the model given in Section 2, it is clear that externality is due to the assumption that the ¯rms ignore the indirect e®ect of an increase in the employment of a factor on function hi(Q 1 ; Q 2 ): As a result, there is a divergence between the private marginal product and social marginal product of a factor.^10 The optimal policy considered here is a set of taxes/subsidies imposed on the employment of factors. We argue that the optimal policy on sector i consists of an
(^10) For more discussion about private marginal product and social marginal product, see Wong (2000a).
ad valorem employment subsidy si plus a lumpsum subsidy of Ái on factor j; where
si =
"ii(1 ¡ "kk) + "ik"ki £
Ái = ¡pi
μ hkFihikFkj £
where i; k = 1; 2, i 6 = k; j = K; L; and the variables are evaluated at the optimal point. Note that a negative subsidy is a tax. To see how this policy works, let us consider the income of labor. The e®ective wage rate consists of three parts: the payment by a ¯rm, the employment subsidy from the government, and the lumpsum subsidy. The ¯rms take the subsidies as given, and keep on ignoring the externality e®ects, i.e., they pay the factors they employ according to their private marginal products. The total income of a worker, which is regarded as the e®ective wage rate, is equal to
w = pi(1 + si)hiFiL + Ái: (55)
To ¯nd out what this wage rate is, we note that
1 + si =
1 ¡ "kk £
Substituting (56) into (55) and using (53) and (54), we have
w = piQiL (57)
for i = 1; 2. Condition (57) means that the wage rate is equal to value of social marginal product of labor. A similar condition can be derived for the rental rate. Thus in the presence of the present policy, both factors are employed optimally.
6 International Trade
We now consider open economies and analyze foreign trade. In order to have mean- ingful comparison of the two countries, we assume that the autarkic equilibrium of each country is unique. As it is done in Wong (2000b), denote the export supply of good 1 of home by E 1 (p; K; L); where p is the domestic price ratio of good 1. Note that both K and L are treated as exogenous. Invert the export supply function E 1 = E 1 (p; K; L) to give p = ½(E 1 ); where for simplicity the factor endowments are
