[, which expresses the commodity price mobile factor employment relationship] cannot both be negative. With

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1 This paper was written on October 6, 206. The enery price commodity output relationship and the commodity price commodity output relationship in a three-factor, two-ood eneral equilibrium trade model with imported enery Yoshiaki Nakada Division of Natural Resource Economics, Graduate School of Ariculture, Kyoto University,.Sakyo, Kyoto, Japan. nakada@kais.kyoto-u.ac.jp Abstract: We analyze how enery price and commodity price affect commodity output in a three-factor, twoood eneral equilibrium trade model with three factors (capital, labor, and imported enery), respectively. We search for a sufficient condition for a specific pattern of each relationship to hold. We assume factor-intensity rankin is constant. We use the EWS (economy-wide substitution)-ratio vector to analyze. We show that the position of the EWS-ratio vector determines the relationships. Specifically, if the price of ood rises, the output of ood 2 can rise, if capital and labor are economywide complements. This article provides a basis for further applications. Keywords: three-factor, two-ood model; eneral equilibrium; the enery price commodity output relationship; the commodity price commodity output relationship; EWS (economy-wide substitution)-ratio vector JEL Classification Numbers: F, F3, F8, Q40, Q48.. Introduction Thompson (983) published an article on functional relations in a three-factor, two-ood neoclassical model (hereinafter 3 x 2 model), where one of factors is internationally mobile. Thompson called it the 3 x 2 mobile factor model. Usin eq. (), Thompson (983, p.47) stated, risin (fallin) payment to the mobile factor cannot increase (decrease) both outputs. And he showed the proof of this. Thompson (983, p.48) continued, Also proven is that v/ p and v/ p2 [, which expresses the commodity price mobile factor employment relationship] cannot both be neative. With p risin, for instance, x rises and x 2 falls. He asserted that if the price of ood rises, the However, I am uncertain whether Thompson (983, p. 47-8) s proof is plausible. Specifically, he used D'( 0) for the proof of the property of eq. (). D ' is the system determinant of the usual 3 x 2 model with all factor payments endoenous, and is known to be neative. However, we don t need it for that proof. Oriinally, he used it for the proof of the property of eq. (9). This is plausible. However, I do not discuss this anymore.

2 output of ood rises and the output of ood 2 falls. In other words, the own price effect on output is positive, and the cross price effect on output is neative. However, Thompson did not show the proof of this. Thompson (994) analyzed the 3 x 2 model with three factors (capital, labor, and imported enery), whose basic structure is essentially the same as shown in Thompson (983). Thompson (994, p.94) called this model the 3 x 2 international enery model. Accordin to Thompson (994, p. 98-9), there are three sin patterns of the enery tariff commodity output relationship. However, he did not show a sufficient condition for a specific sin of that relationship to hold. Moreover, Thompson (2000) analyzed the same model. He analyzed the effect of enery tax on wae. Thompson (204) analyzed a 3 x 2 model with three factors (capital, labor, and imported enery) in elasticity terms, whereas Thompson (983, 994, 2000) analyzed in differential forms. The model in Thompson (204) explicitly included tariff revenue and (national) income. For example, Thompson (204) analyzed how the enery tariff affected the enery import, commodity output, and domestic factor price. He also analyzed how capital, labor, and commodity price affected the (national) income includin tariff revenue. Moreover, he did a simulation study under the assumption of Cobb Doulas production functions. Thompson (204, p.65) stated, Factor intensity and substitution both play roles in output adjustments. The [enery] tariff must lower at least one of the two outputs as shown by Thompson (983). Thompson (204) implicitly assume enery price abroad is constant. On this, see Appendix. In summary, the models in Thompson (983, 994, 2000, 204) are the 3 x 2 models, where one of factor payments is exoenous. However, for example, the equation shown in Thompson (204), which expresses the enery tariff commodity output relationship included an error. On this, see Appendix. Moreover, Thompson (204) did not mention about the price effect on output mentioned in Thompson (983). I am uncertain whether all of Thompson s results are plausible. We can show that the enery tariff commodity output relationship in Thompson (204) is similar to the enery price commodity output relationship in a 3 x 2 model with no enery tariff shown in this article (see eq. (27)). (i) The followin questions arise. Can we derive a sufficient condition for a specific sin of the enery price commodity output relationship to hold? (ii) Is Thompson (983) s statement plausible that if the price of ood rises, the output of ood rises and the output of ood 2 falls? To the best of my knowlede, other than Thompson (983, 994, 2000, 204), no studies analyzed a 3 x 2 model, where one of factor payments is exoenous. 2 We can classify other studies 2 For example, we can make a 3 x 2 trade model with three factors (capital, labor, and 2

3 that analyzed the model with one of factor payments exoenous as follows. (i) Studies that analyzed a two-factor, two-ood, two-country model (or 2 x 2 x 2 model), where one factor, capital, is internationally mobile. For example, Mundell (957, p. 322), Kemp (966), Jones (967), Chipman (97), Jones and Ruffin (975), and Feruson (978, p. 374). (ii) Studies that analyzed a 2 x 2 x 2 model, where each sector employs domestic labor and reenhouse-as emission traded internationally. For example, Ishikawa and Kiyono (2006, p ), and Ishikawa et.al (202, p. 87-8). (iii) Studies that analyzed the simplest type of a 3 x 2 model, what you call, specific factors model, where the factor specific to one sector is internationally mobile. For example, Srinivasan (983, p. 292), Brecher and Findlay (983), Thompson (985), Dei (985), Bandyopadhyay and Bandyopadhyay (989), and Oino (990). (iv) Studies that analyzed a N-factor, M-ood model, where some factors are internationally mobile. For example, Svensson (984) and Ethier and Svensson (986, p. 22). Ethier and Svensson (986) examined some theorems such as the Rybczynski and Stolper-Samuelson theorems. In this article, we do a detailed analysis on functional relations in a 3 x 2 model with three factors (capital, labor, and imported enery). We do not include the enery tariff into the model for ease of analysis. The purposes of this article are as follows. First, we derive a sufficient condition for a specific sin of the enery price commodity output relationship to hold. We can easily show that this relationship is dual counterpart in the commodity price enery import relationship. We also analyze the latter. Second, we analyze the commodity price commodity output relationship. We prove (or disprove) the Thompson (983) s statement about the price effect on output. We assume factor-intensity rankin is constant. Likely to Thompson (204, eq. (9)), we environmental tax which is exoenous). The studies shown below look like such a model, however, they are not. Takeda (2005, p. -2) used a model with two oods where each sector employs three factors (capital, labor, and emission tax which is specific to each sector). Hence, his model is a fourfactor two-ood trade model, where two of factor payments are specific and exoenous. However, Ishikawa and Kiyono (2006, p. 432, note 3) stated, Takeda (2005) had extended our model [2 x 2 x 2 model] to a two-ood, three-factor model. Takeda (2005, p. ) stated, Thus, the model has a structure similar to the standard 2 3 [3 x 2 in our expression] HO [or Heckscher-Ohlin] model employed in Batra and Casas (976) and Jones and Easton (983). This expression is confusin. On the other hand, Heutel and Fullerton (200, p. 3-4) used a 3 x 2 model where pollution tax (tax or other price on emissions) is specific to the dirty sector. In their model, commodity price is endoenous. Hence, this model is not a 3 x 2 trade model. For details of their model, see Fullerton and Heutel (2007, p ). 3

4 assume that sector is relatively enery-intensive, and sector 2 is relatively labor-intensive, capital is the middle factor, and enery and labor are extreme factors. We also assume factor-intensity rankin of the middle factor is constant. 3 That is, we assumed the middle factor is used relatively intensively in sector. We analyze in elasticity terms. Batra and Casas (976) (hereinafter BC) published an article on functional relations in a 3 x 2 model with all factor payments endoenous. Nakada (206a) defined the EWS-ratio vector based on economy-wide substitution (hereinafter EWS) oriinally defined by Jones and Easton (983) (hereinafter JE) and used it in an analysis of a 3 x 2 model of BC s oriinal type. By movin some terms in the basic equations in this 3 x 2 model, we derive the basic equations in a 3 x 2 model with imported enery. In this article, we also use the EWS-ratio vector for the analysis. Section 2 of this study explains the model. In section 2., we explain the basic structure of the model. We make a system of linear equations usin a 5 x 5 matrix. In section 2.2, we assume factor-intensity rankin. In section 2.3, we define the EWS-ratio vector based on EWS for the analysis. We derive the important relationship amon EWS-ratios and draw the EWS-ratio vector boundary in the fiure, which is useful for our analysis. In section 2.4, we derive the solutions of a system of linear equations. In section 2.5, we analyze the enery price commodity output relationship usin EWSratios. We draw the border line for a sin pattern of that relationship to chane in the fiure, which we call line Tj. Line Tj divides the reion of the EWS-ratio vector into four subreions. We analyze sin patterns usin the Hadamard product of vectors and derive a sufficient condition for a specific sin pattern to hold. In section 2.6, we analyze the commodity price enery import relationship. In section 2.7, we analyze the commodity price commodity output relationship. Section 3 presents the conclusion, and the Appendix shows that some equations in Thompson (204) include errors. Section 2.3 contains similar content as Nakada (206a). 2. Model 2.. Basic structure of the model We assume similarly to Thompson (204). However, we do not include enery tariff. We modify the assumptions in a 3 x 2 model in BC (p. 22-3) and Nakada (206a). In Nakada (206a), we assumed as follows. Products and factor markets are perfectly competitive. The supply of all factors is perfectly inelastic. Production functions are homoeneous of deree one and strictly quasi-concave. All factors are not specific and perfectly mobile between sectors, and factor prices are perfectly 3 Thompson (994, 2000) assumed factor-intensity rankin differently. He assumed that sector is relatively enery-intensive, and sector 2 is relatively capital-intensive. Thompson (983, 994, 2000, 204) did not assume about factor-intensity rankin of the middle factor. 4

5 flexible. These two assumptions ensure the full employment of all resources. The country is small and faces exoenously iven world prices, or the movement in the price of a commodity is exoenously determined. The movements in factor endowments are exoenously determined. In a 3 x 2 model in Nakada (206a), three factors are land, capital, and labor. In a 3 x 2 model in this article, three factors are capital, and labor, and imported enery. Specifically, in Nakada (206a), w T and V T denote the reward for land and land endowment, respectively. In this article, these symbols denote the enery price and the amount of imported enery, respectively. They are exoenous and endoenous variables, respectively. Full employment of factors implies a X V j, i T i, K, L, () j where X j denotes the amount produced of ood j (j = l, 2); a denotes the requirement of input i per unit of output of ood j (or the input-output coefficient); V i denotes the supply of factor i; T is the enery, K capital, and L labor. In a perfectly competitive economy, the unit cost of production of each ood must just equal its price. Hence, aw i pj, j,2, (2) i where p j is the price of ood j, and w i is the reward of factor i. BC (p. 23) stated, With quasi-concave and linearly homoeneous production functions, each input-output coefficient is independent of the scale of output and is a function solely of input prices: a a w, i T, K, L, j,2. (3) i The authors continue, In particular, each C [a in our expression] is homoeneous of deree zero in all input prices. 4 Equations ()-(3) describe the production side of the model. These are equivalent to eqs ()-(5) in BC. The set includes equations in endoenous variables (X j, a, w k, w L, and V T) and 4 From the condition of cost minimization, we can show that a is homoeneous of deree zero in all input prices (see Samuelson (953, chapter 4, p. 68), Nakada (206a, p. 5)). 5

6 five exoenous variables ( VK, VL, w T, and p j ). The small country assumption simplifies the demand side of the economy. Totally differentiate eq. (): j ( a * X *) V *, i T, K, L, (4) j i where an asterisk denotes a rate of chane (e.., X * d X / X j j j ), and where λ is the proportion of the total supply of factor i in sector j (that is, a X / V ). Note that. j i j The minimum unit cost equilibrium condition in each sector implies w da 0. Hence, we derive (see JE (p. 73), BC (p. 24, note 5), i i a * 0, j, 2, (5) i where θ is the distributive share of factor i in sector j (that is, a w / p ). Note that ; da is the differential of a. Totally differentiate eq. (2): i j i w * p *, j, 2. (6) i i j Totally differentiate eq. (3) to obtain a * w * 0, i T, K, L, j, 2, (7) h h h where loa / lo w. (8) h h hj h h is the AES (or the Allen-partial elasticities of substitution) between the ith and the hth factors in the jth industry. For an additional definition of these symbols, see Sato and Koizumi (973, p. 47-9), BC (p. 24). AESs are symmetric in the sense that h hj. (9) i 6

7 Accordin to BC (p. 33), Given the assumption that production functions are strictly quasi-concave and linearly homoeneous, 0. (0) i Because a is homoeneous of deree zero in all input prices, we have h h hhj h 0, i T, K, L, j, 2. () Eqs (7) to () are equivalent to the expressions in BC (p. 24, n.6). See also JE (p. 74, eqs (2)-(3)). Substitutin eq. (7) in (4), we derive ( w * X *) w * X * V i*, T, K L, (2). j h h h j h ih h j j i where, i, h T, K, L. (3) ih j h This is the EWS (or the economy-wide substitution) between factors i and h defined by JE (p. 75). ih is the areate of h. JE (p. 75) stated, Clearly, the substitution terms in the two industries are always averaed toether. With this in mind, we define the term i k to denote the economy-wide substitution towards or away from the use of factor i when the kth factor becomes more expensive under the assumption that each industry's output is kept constant. We can easily show that hih 0, i T, K, L, (4) ih ( h / i) hi, i, h T, K, L, (5) where i and ross national income. That is, j pjxj / I, i wv i i / j are, respectively, the share of factor i, i T, K, L, and ood j, j,2 in I, where I jpjxj iwiv. See BC (p. i 25, eq. (6)). 5 Hence, we obtain ( j / i) (see JE (p. 72, n. 9)). Note that j j, i i. ih is not symmetric. Specifically, ih hi, i h in eneral. On eq. (5), see also JE (p. 85). From eqs (8), (0), and (3), we can show that 5 Note that in a 3 x 2 model with all factor payments endoenous, BC called I national income, and Nakada (206a) called I total income. On the other hand, in a 3 x 2 model in this article, we define the national income as follows. I ' I wtvt. Thompson (204) did not use i and j. 7

8 0. (6) ii From eqs (4) and (6), we derive KT KL KK 0, TK TL TT 0, LK LT LL 0. (7) From eqs (5) and (7), we can easily show that (,, ) LK LT KT,,,,,,,,,,,. (8) = At most, one of the EWSs ( LK, LT, KT ) can be neative. Movin the term of w T * in eqs (6) and (2) to the riht hand side, and the term of V T * in eq. (2) to the left hand side, make new equations. Next, combine these equations to make a system of linear equations. Usin a 5 x 5 matrix, we obtain where A= AX = P, (9) 0 K L K2 L2 0 0 TK KK LK TL KL LL T T2 K K2 A is a 5 x 5 coefficient matrix, and X, P are column vectors. L L2 VT * p* TwT* wk * p2* T2w * T, X= wl *, P= TTwT *. X * K V * KTwT * X 2* VL * LTwT * 2.2. Factor-intensity rankin In this article, we assume T K L, (20) T 2 K 2 L2. (2) K K2 Eq. (20) is, what you call, the factor-intensity rankin (see JE (p. 69), see also BC (p. 26-7), Suzuki (983, p. 42)). 6 This implies that sector is relatively enery-intensive, and sector 2 is relatively labor-intensive, capital is the middle factor, and enery and labor are extreme factors (see also Ruffin (98, p. 80)). Eq. (2) is the factor-intensity rankin for middle factor (see JE (p. 70)). It implies 6 For the details of the factor-intensity rankin, see Nakada (206a). 8

9 that the middle factor is used relatively intensively in sector. Define that ABE, T T 2, K K L, (, ). (22) 2 L2 This is the inter-sectoral difference in the distributional share. Recall (5) ( ), which implies A B E 0. (23) Because we assume eqs (20) and (2) hold, we derive i A, B E (,,, ). (24) 2.3. EWS-ratio vector boundary In this section, we show the important relationship between EWS-ratios, and we draw the EWS-ratio vector boundary in the fiure. This is useful for our analysis. Each a function is homoeneous of deree zero in all input prices (see eq. (3)). Accordin to BC (p. 33), Given the assumption that production functions are strictly quasi-concave and linearly homoeneous, 0 (see eq. (0)). This implies (see eq. (39) in Nakada (206a)) i L S' L S' U ', if T 0; U ' ' ', ift 0, (25) K S K S where S, U S / T, U / T /, /, (26) S T U LK LT KT LT,,,,. (27) LK LT KT We call S, U the EWS-ratio vector. S denotes the relative manitude of EWS between factors L and K compared to EWS between factors L and T. U denotes the relative manitude of EWS between factors K and T compared to EWS between factors L and T. Transform 9

10 L S ' L L U ' K S' K K S', (28) which expresses the rectanular hyperbola. We call this the equation for the EWS-ratio vector boundary. It passes on the oriin of O (0, 0). The asymptotic lines are S U' L/ K,. We can draw this boundary in the fiure (see Fi. ). S is written alon the horizontal axis and U alon the vertical axis. The EWS-ratio vector boundary demarcates the boundary of the reion for the EWS-ratio vector. This implies that the EWS-ratio vector is not so arbitrary, but exists within these bounds. Note that: The EWS-ratio vector (S', U ) exists in the upper-riht reion of the EWS-ratio vector boundary, if T > 0, The EWS-ratio vector exists in the lower-left reion of the EWS-ratio vector boundary, if T < 0. (29) The sin pattern of the EWS-ratio vector is, in each quadrant (on this, see also eq. (8)): Quad. I: S U S T U ',,,,,, ; Quad. II: S U S T U ',,,,,, ; Quad. III: S U S T U ',,,,,, ; Quad. IV: S U S T U ',,,,,,. (30) Hence, one of the EWS can be neative at most. Note that T > 0, if (S', U ) exists in quadrant I, II, or IV, T < 0, if (S', U ) exists in quadrant III, (3) We define (for i h), Factors i and h are economy-wide substitutes, if ih 0, Factors i and h are economy-wide complements, if ih 0. (32) 2.4. Solution Usin Cramer s rule to solve eq. (9) for X *, we derive X * 4 /, (33) 0

11 where = det (A), det 0 K L p* TwT * 0 0 K 2 L2 p2* T 2wT * 0 4 A 4 TK TL TTwT * T 2. 0 KK KL VK * KTwT * 0 V * w * K 2 LK LL L LT T L2 is the determinant of matrix A. Replacin column 4 of matrix A with column vector P, we derive the matrix A 4. 4 is the determinant of matrix A 4. Express the above as a cofactor expansion alon the first column: K L K 2 L2 0 0 K L ( ) ( ) K K2 KK KL K K 2 K 2 L2 L L2 LK LL L K L K L ( K L2 K 2 L) 0. K 2 L2 K 2 L2 K L K L L2 (34) K L p* TwT * 0 3 K 2 L2 p2* T 2wT * 0 4 ( ) ( )[ J ( wt*) K ], (35) KK KL VK * KTwT * K 2 V * w LK LL L LT T * L2 where J K L p* 0 K L T 0 K 2 L2 p2* 0 K 2 L2 T 2 0, K, (36) KK KL VK * K 2 KK KL KT K 2 V * LK LL L L2 LK LL LT L2 Express the above as a cofactor expansion alon the third column: J K K L CL p * C p * C V * C V *, (37) where

12 K 2 L2 0 K L 0 K L 0 K L 0 C, C, C 0, C 0, KK KL K2 2 KK KL K2 K K 2 L2 L K 2 L2 LK LL L2 LK LL L2 LK LL L2 KK KL K2 (38) Sum columns and 2 in column 3, and subtract row 2 from row. Next, express as a cofactor expansion alon the third column. We have K B E L2 2 3 K2 L2 () CT, (39) KK KL K 2 LK LL 0 0 ABE,, (, ), and where, L where recall eq. (22), that is, T T2 K K2 L2 C B E 0 T KK KL K 2. (40) LK LL L2 From eqs (33), (35), (37), and (39), we have X * / ( )[ J ( wt*) K ] 4 p* C p2*( C V * C V *( C wt C ) ) ( *)( ). 2 K K L L T (4) On the other hand, solve eq. (9) for X 2* : X 2* 5 /, (42) where det A K L 0 p* TwT * 0 K 2 L2 0 p2* T 2wT * TK TL T TTwT * 0 KK KL K VK * KTwT * 0 LK LL L VL * LTwT *. Replacin column 5 of matrix A with column vector P, we derive matrix A 5. 5 is the determinant of matrix A 5. Express the above as a cofactor expansion alon the first column: 2

13 K L 0 p* TwT * 3 K 2 L2 0 p2* T 2wT * 5 ( )[ L ( wt*) M ], (43) KK KL K VK * KTwT * LK LL L VL * LTwT * where L 0 p * 0 K L K L T K 2 L2 0 p2* K 2 L2 0 T 2, M. (44) KK KL K VK * KK KL K KT V * LK LL L L LK LL L LT Express the above as a cofactor expansion alon the fourth column: L K K 2 L CL2 p * C p * C V * C V *, (45) where K 2 L2 0 K L 0 K L 0 K L 0 C, C, C 0, C 0. 2 KK KL K 22 KK KL K K2 K 2 L2 L2 K 2 L2 LK LL L LK LL L LK LL L KK KL K (46) Sum columns and 2 in column 4, and subtract row 2 from row. Next, express as a cofactor expansion alon the fourth column. We have M B E 0 0 K2 L () CT 2, (47) KK KL K LK LL L 0 0 ABE,, (, ), and where, L where recall eq. (22), that is, T T2 K K2 L2 C B E 0 T 2 KK KL K. (48) LK LL L Hence, from eqs (42), (43), (45), and (47), we have 3

14 X * / ( )[ L ( wt*) M ] 2 5 p* ( C2 ) p2* C2 2 VK *( CK 2) VL * CL2 ( wt*) CT2. (49) In summary, from eqs (4) and (49), we obtain X* p* C p2*( C2) VK* CK VL *( CL ) wt * CT, X 2* p* ( C2 ) p2* C22 VK *( CK 2) VL * CL2 wt* CT2. (50) (5) From eqs (50) and (5), we have X */ wt * CT CT ( ), X */ w * C C 2 T T 2 T 2 (52) X */ p* X */ p2* C ( ) C2. X 2*/ p* X 2*/ p2* C2 C 22 (53) Equations (52) and (53) express the enery price commodity output relationship and the commodity price commodity output relationship, respectively Enery price commodity output relationship We analyze the enery price commodity output relationship. From eq. (7), we derive ( ) and ( LK LT). Substitute these equations in eq. (40) and transform. KK KT KL We have LL 4

15 B E 0 B E 0 0 E 0 C T KK KL K 2 KL KL K 2 KT KL K 2 0 LK LL L2 LK LL L2 LL L2 B E 0 B E( ) KL KL K 2 KL K 2 KT L2 LK LK L2 LK LT L2 B E B E K 2 L2 B( LT) K 2 E( KT) L2 LK LK KL KL K 2( B E) LK L2( B E) KL BK2LT EL2KT. (54) Usin eq. (5), eliminate KL from (54). We derive 2 CT ( B E)( K 2 L2) LK BK2LT EL2KT. (55) K Similarly from eq. (48), we derive CT 2 ( B E)( K L ) LK BKLT ELKT. (56) K From eq. (23), we have B E A. Substitutin this in eqs (55) and (56). Recall eq. (27), that is,s, T, U,, (56): LK LT KT 2 CT ( A)( T 2) S B K2T EL2U, K. Usin these symbols for ease of notation, transform eqs (55) and CT 2 ( A)( T) S B KT ELU. (57) K Substitutin eq. (57) in (52), we derive 2 X */ wt* ( CT) ( A)( T 2) S B K 2T EL2U, K X 2*/ wt* CT2 ( A)( T) S BKT ELU. K However, eq. (58) is not equivalent to the equation which Thompson (204, eq. (5)) showed. Thompson s equation includes an error. On this, see Appendix. (58) 5

16 C T and T 2 C is a linear function in S, T, and U. Recall eq. (26), that is, S, U S / T, U / T /, / transform eq. (57) to derive LK LT KT LT ' E U S ', which we call the EWS-ratio vector. Usin this, CT EL2T[ U ' ft S ], CT 2 LT [ ' ft2 ], (59) where 2 fts' [ A( T 2) S ' B K 2]( EL2), K ft 2S' [ A( T) S ' B K]( EL). (60) K Substitutin eq. (59) in (52), we derive L2 U ft ' X */ wt* ( CT ) E T S [ ' ], X 2*/ wt* CT2 ELT [ U ' ft2s ' ]. (6) Usin the Hadamard product of vectors, we can transform eq. (6): X */ wt * EL2 U ' ft S T. X */ w * E U ' f S 2 T L T 2 ' ' (62) In eneral, if A = [a ] and B = [b ] are each m x n matrices, their Hadamard product is the matrix of element-wise products, that is, A B [ab ]. For this definition, see, for example, Styan (973, p. 27-8). Hadamard product is known, for example, in the literature of statistics. The sin pattern eq. (62) is expressed: X */ wt * EL2 U ' ft S' sin sin T. X 2*/ wt * E L U ' ft 2 S' (63) From eq. (6), we derive Xj*/ wt* 0 U ' f S ', j,2. (64) Tj This equation expresses the straiht line in two dimensions. We call it the equation for line Tj, which 6

17 expresses the border line for a sin pattern of Xj*/ w T* to chane. Usin eqs (64) and (28), make a system of equations: U' f S', j,2, (65) Tj L U ' K S ' S '. (66) From these, we obtain a quadratic equation in S for each j. Solve this to derive two solutions. Each solution denotes the S coordinate value of the intersection point of line Tj and the EWS-ratio vector boundary. The solutions are, for line-t and T2, respectively: S B A, K 2, T 2 S B A, K. (67) T In summary, there are three intersection points. Each line Tj passes throuh the same point, which we call point Q. The Cartesian coordinates of point Q are B B,,. (68) A E K L S U ( ) We call two intersection points other than point Q, the point R, j, 2. The Cartesian coordinates of these points are, for line-t and T2, respectively: Tj K K L K K L S, U (, ),(, ) T 2 L2 K T L K 2 2. (69) From eqs (20) and (2), we derive eq. (24), that is, A, B, E,,. Substitutin this in eq. (68), we derive the sin pattern of point Q, that is, sin S, U,. (70) This implies that point Q belons to quadrant IV. The sin patterns of point R T and R T 2 are, respectively, sin S, U,,,. (7) 7

18 Hence, both of points R T and R T2 are in quadrant II. Next, we investiate the relative position of points R T and R T2. From eq. (20), we can prove for S values of R T and R T2: K2 K T 2 T ( K 2 K L L K2 L2 Equation (72) explains the relative position of these two points. ) 0. (72) From eqs (68)-(72), we can draw point Q and R Tj and, hence, line Tj in the fiure. Each line Tj divides the reion of the EWS-ratio vector into 4 subreions, that is, the subreion P-3 (upperriht reion) and M (lower-left reion) (see Fi. ). In eneral, if the EWS-ratio vector (S, U ) exists in the subreion above line Tj (resp. below line Tj), we derive f S resp U f S U ' Tj 0 (. ' Tj 0) Sin patterns of vector U ' f S Tj. (73) are, respectively, for each subreion: P P2 P3 M si U ' ft S n U T 2 ' ',,,. ' f S (74) Of course, we can state that from eq. (3) and Fi., T > 0, if the EWS-ratio vector exists in any of the subreions P-P3, T < 0, if the EWS-ratio vector exists in the subreion M. (75) Recall that 0. (see eq. (34)), and recall that we assume eq. (24), that is, ( A, B, E) (,, ). Substitutin these in eq. (63), we have X */ w T * ' ft S sin s n T U i. X */ w * U ' f S ' ' 2 T T 2 (76) From eqs (74)-(76), we can state as follows. The sin patterns of Xj*/ wt* for the subreions P- P3; M are: 8

19 P P2 P3 M X*/ wt * sin X 2 w T,,,. */ * (77) There are three patterns in total. Therefore, we can make the followin statements. (i) If the EWS ratio vector (S, U ) exists in subreion P, the effects of enery price on commodity output in sector and sector 2 are positive and neative, respectively. (ii) If the EWS ratio vector exists in subreion P2, both the effects of enery price on commodity output in both sectors and 2 are neative. (iii) If the EWS ratio vector exists in subreion P3, the effects of enery price on commodity output in sector and sector 2 are neative and positive, respectively. (iv) If the EWS ratio vector exists in subreion M, the effects of enery price on commodity output in sector and sector 2 are neative and positive, respectively. Here we assume S', U,. The EWS-ratio vector exists in quadrant III, that is, in the subreion M. From eq. (30), we derive S, U S / T, U / T, S T U,,,,. LK, LT, KT (78) This implies that labor and enery, extreme factors, are economy-wide complements. From eq. (77), we derive X*/ wt * sin X 2 w T. */ * The followin result has been established. Theorem. We assume the factor intensity rankin as follows. T K L, (79) T 2 K 2 L2. (80) K K2 Further, if the EWS-ratio vector S, U exists in quadrant III (or subreion M), in other words, if enery and labor, extreme factors, are economy-wide complements, the effects of enery price on 9

20 commodity output in sector and sector 2 are neative and positive, respectively. X*/ wt * sin X 2 w T. */ * (8) 2.6. The commodity price enery import relationship. In this section, we analyze the commodity price enery import relationship. We define the national income as follows (see footnote 5). I ' I wtvt px p2x 2 wtvt. (82) Recall eq. (5), that is, I jpjxj iwiv. I is the ross national income. We use a similar way i to BC (p. 36) who used the reciprocity relations derived by Samuelson. In particular, we derive I ' Xj, pj I ' wt VT. (83) It follows that or I ' I ' Xj ( V T), pj pj wt wt pj wt (84) VT * j Xj *, j,2. pj * T wt * (85) Hence, the commodity price enery import relationship is dual counterpart in the enery price commodity output relationship. Equation (84) is equivalent to the part of equation which Thompson (983, eq. ()) showed. However, he did not show the proof. Substitutin eq. (58) in (85), we derive VT * p* 2 ( A)( T 2) S B K2T EL2U T K VT * 2 ( A)( T) S B KT ELU p2* T K. (86), However, eq. (86) is not equivalent to the equation which Thompson (204, eq. ()) showed. On this, see Appendix. 20

21 From eqs (85) and (77), the sin patterns of VT*/ pj* for each subreion: P P2 P3 M VT */ p* sin V T p 2,,,. */ * (87) There are three patterns in total. For example, we can make the followin statements. (i) If the EWS ratio vector (S, U ) exists in subreion P, the effects of the price of commodity and commodity 2 on enery import are neative and positive, respectively. (ii) If the EWS ratio vector (S, U ) exists in subreion P2, both the effects of the price of commodity and commodity 2 on enery import are positive. (iii) If the EWS ratio vector (S, U ) exists in subreion P3, the effects of the price of commodity and commodity 2 on enery import are positive and neative, respectively. (iv) If the EWS ratio vector (S, U ) exists in subreion P4, the effects of the price of commodity and commodity 2 on enery import are positive and neative, respectively Commodity price commodity output relationship We analyze the commodity price commodity output relationship. Usin a similar way to transform C T in eq. (40) (see eqs (54)-(55)), expand eqs (38) and (46). We derive 2 C ( K 2 L2)( K 2 L2) LK K 2K2LT L2L2KT, (88) K 2 C2 ( K L )( K 2 L2) LK K K2LT L L2KT, (89) K C2 ( K 2 L2)( KL ) LK K 2KLT L2LKT, (90) K C22 ( K L )( KL ) LK K KLT L LKT. (9) K Recall eq. (27), that is, S, T, U,,. Usin these symbols, transform eq. (89): LK LT KT C as bt c U, (92) 2 where 2 a ( )( ), b K2, c L2. K L K 2 L2 K L K 2

22 C 2 is a linear function in S, T, and U. Recall eq. (26), that is, S, U S / T, U / T /, / LK LT KT LT. Usin this, transform eq. (92) to derive 2 2 C ct U f S, (93) where a b f2 S S. (94) c c From eqs (53) and (93), we derive C [ U ' f2 S ]. (95) X */ p2* ( 2) ct The sin pattern of eq. (95) is expressed: sin[ X */ p2*] sin[ ct[ U' f2 S ]]. (96) From eq. (95), we derive 2 2 ' a */ * 0 b X p U f S S. (97) c c This equation expresses the straiht line in two dimensions. We call it the equation for line 2, which expresses the border line for a sin of X*/ p 2* to chane. The radient and intercept of line 2 are, respectively, a/ c and b/ c. Usin eqs. (97) and (28), make a system of equations: a b U f2 S S, (98) c c L U ' K S ' S '. (99) From eqs (98) and (99), we obtain a quadratic equation in S : a 2 a b L b ( S ) ( ) S ' 0. (00) c c c K c We define for ease of notation: x /, y /, z /. (0) K L L K K 2 L2 22

23 We derive a b a b L ( x)( z) y, xzy, [ ( x z) 2xz] y. (02) c c c c K Substitute eq. (02) in (00), and divide the both sides by y. We derive: 2 ( x)( z)( S [( x ] S ' ') z) 2xz xz 0. (03) The discriminant of eq. (03) is 2 D [( x z) 2 xz] 4( x)( z) xz ( xz ) ( K K K K ) ( L ) ( 0). L L2 L K 2 L2 (04) From eq. (20), we derive D 0. Hence, eq. (03) has two distinct real solutions. This solution denotes the S coordinate value of the intersection point of line 2 and the EWS-ratio vector boundary. The solutions are: S z x z x, K2, K. T2 T (05) The Cartesian coordinates of the intersection point of line 2 and the EWS-ratio vector boundary are K K L K K L S, U (, ),(, ). T 2 L2 K T L K 2 2 (06) These points are the same as the points R T and R T 2 mentioned above, respectively (see eq. (69)). Hence, we can draw line 2 in the fiure. Line 2 divides the reion of the EWS-ratio vector into three subreions, that is, the subreion Pa-b (upper-riht reion) and Ma (lower-left reion) (see Fi. 2). we derive If the EWS-ratio vector (S, U ) exists in the subreion above line 2 (resp. below line 2), f S ( resp U ' f S U ' 0 2 Sin patterns of U ' f S. 0) are, respectively, for each subreion: 2. (07) 2 Pa Pb Ma 23

24 sin[ U ' f S ] ( ),( ),( ). (08) 2 Of course, we can state that from eq. (3) and Fi. 2, T > 0, if the EWS-ratio vector exists in any of the subreions Pa-Pb, T < 0, if the EWS-ratio vector exists in the subreion Ma. (09) Recall that 0. (see eq. (34)). Substitutin this and eqs (08), (09) in (96), the sin patterns for each subreion are: Pa Pb Ma sin[ X */ p2*] ( ),( ),( ). (0) Usin eq. (83), we derive Xj I ' I ' Xs, j, s,2, j s. ps ps pj pj ps pj () From eqs (0) and (), the sin patterns of X 2*/ p * for the subreion Pa, Pb, Ma are: Pa Pb Ma sin[ X 2*/ p*] ( ),( ),( ). (2) (i) (ii) (iii) From eq. (2), we can make the followin statements. If the EWS ratio vector (S, U ) exists in subreion Pa, the effect of price of commodity on output of commodity 2 is positive. If the EWS ratio vector exists in subreion Pb, the effect of price of commodity on output of commodity 2 is neative. If the EWS ratio vector exists in subreion Ma, the effect of price of commodity on output of commodity 2 is neative. (i) is aainst Thompson s (983, p. 48) statement that if the price of ood rises, the output of ood 2 falls. From eq. (72), factor intensity rankin is important only to decide the relative position of points R T and R T2. Hence, without reard to the factor intensity rankin, subreion Pa exits in quadrant II, in which capital and labor are economy-wide complements. 24

25 Hence, the followin result has been established. Theorem 2 If capital and labor, domestic factors, are economy-wide complements, the effect of price of commodity on output of commodity 2 can be positive, without reard to the factor intensity rankin. In eneral, this can occur in a three-factor, two-ood neoclassical model, where one of factor payments is exoenous. We can analyze how the strenthenin (or reduction) of import restrictions affected the commodity output of exportables. Let industries and 2, respectively, sinify exportables and importables. We may conclude as follows. If capital and labor, domestic factors, are economy-wide complements, the strenthenin of an import restriction which raises the domestic price of importables can increase the commodity output of exportables. Similarly, the reduction of import restrictions which decreases the domestic price of importables can decrease the commodity output of exportables. On the other hand, we can show that sin[ Xj*/ pj*] ( ), j,2. (3) I omit the proof. Hence, if the price of ood j rises, the output of ood j rises. This is not aainst Thompson s (983, p. 48) statement that if the price of ood rises, the output of ood rises. 3. Conclusion We assumed a certain pattern of factor-intensity rankin, includin a certain pattern of factor-intensity rankin of the middle factor. We assumed sector is relatively enery-intensive, and sector 2 is relatively labor-intensive, capital is the middle factor, and enery and labor are extreme factors. Further, we assumed the middle factor is used relatively intensively in sector. (iii) (iv) In the introduction, we posed the followin questions. Can we derive a sufficient condition for a specific sin of the enery price commodity output relationship to hold? Is Thompson (983) s statement plausible that if the price of ood rises, the output of ood rises and the output of ood 2 falls? We derive the results as follows. Answer to (i): We analyzed the enery price commodity output relationship usin the EWSratio vector. The EWS-ratio vector boundary demarcates the boundary of the reion where the EWSratio vector can exist. Line Tj divides this reion into four subreions. There are three patterns of the 25

26 enery price commodity output relationship. The position of the EWS-ratio vector determines that relationship. We derived a sufficient condition for a specific relationship to hold. Notably, if the EWSratio vector S, U exists in quadrant III (or subreions M), in other words, if enery and labor, extreme factors, are economy-wide complements, a specific result holds necessarily (see eq. (8)). We also analyzed the commodity price enery import relationship, which is dual counterpart in the enery price commodity output relationship. Answer to (ii): We analyzed the commodity price commodity output relationship. Line 2 divides the reion where the EWS-ratio vector can exist into three subreions. Notably, if the EWS ratio vector exists in subreion Pa, the effect of price of commodity on output of commodity 2 is positive. This implies that the cross price effect on output can be positive, if the EWS-ratio vector exists in quadrant II, in other words, if capital and labor, domestic factors, are economy-wide complements. And that effect can occur without reard to the factor intensity rankin. This is aainst Thompson s (983, p. 48) statement that if the price of ood rises, the output of ood 2 falls. In eneral, this can occur in a three-factor, two-ood neoclassical model with one factor payment exoenous (e.., enery price, enery tariff, carbon tax, or reenhouse-as emission traded internationally). For example, this suests that free trade areement affects both the outputs of exportables and importables neatively in some cases. This seems paradoxical. In eneral, we can expect that the positive cross price effect on output mentioned above will occur in a N-factor two-ood model, where some factor payments are exoenous, if some pairs of factors are economy-wide complements. Therefore, this article miht be suestive for researchers who do a simulation study in CGE (or computable eneral equilibrium) model. Can we estimate the position of the EWS-ratio vector? As Nakada (206a) has shown in the 3 x 2 model of BC s oriinal type, the EWS-ratio vector exists on the line sement. 7 Even in a 3 x 2 model with imported enery, the EWS-ratio vector exists on the line sement. Therefore, we can estimate the position of the EWS-ratio vector. This article provided the basis for such applications. This article will be useful for efforts to derive the enery price commodity output relationship and the commodity price commodity output relationship in some countries. For example, this study contributes to international, development and enery economics. Appendix: We show that Thompson (204) s equations include errors. By replacin some symbols in 7 Usin this relationship, the author developed a method to estimate the position of the EWS-ratio vector. That is, we can estimate it to some extent, if we have the appropriate data. Further, Nakada (206c) applied Nakada s (206b) results to data from Thailand and, in doin so, derived the factor endowment commodity output relationship for Thailand durin the period

27 Thompson (204), we can easily compare Thompson s equations with ours. Thompson (204, eqs (4), (5), and (9)) defined as follows. ' dt / ( t), KE Kj( akj '/ '), (4) EK E K 2 E2 K EK E K 2 E2 K j,. (5) Prime ' denotes a percentae chane. t denotes the enery tariff. The domestic enery price ed e D. t e chanes with the tariff accordin to ded edt. ' is the percentae chane in KE is an example of the elasticities of input substitution (or EWS in our terminoloy). E is the enery, K capital, and L labor. Thompson (204) called and the factor shares of revenue and the industry employment shares, respectively. The definition of the same as ours. and by Thompson are Note that Thompson (204) implicitly assumed enery price abroad was constant. If we assume enery price abroad ( e ) is variable, we derive e D ded d[( t) e] d( t) de ' ' e'. ed ed t e (6) If we use our expression, eq. (6) reduces: ed* * wt *. (7) In our model, we assume * 0, and in Thompson (204) s model, he assumed wt* 0. Thompson (204, eq. (5)) showed the equation, which expresses the enery tariff commodity output relationship: X '/ ' ( K2 3 L2 4) /, (8) X 2'/ ' ( K3 L 4) /, (9) where X j is the output of commodity j, is the determinant of the system of equations, 3 EL LK ( KL EK ) EL EK KL, 4 ( KL EL) EK KL LK EK KL. (20) Replacin E and ih in eqs (5) and (20) with T and ih, respectively, we 27

28 derive,, (2) TK T K 2 T 2 K TK T K 2 T 2 K ( ), ( ). (22) 3 TL LK KL TK TL TK KL 4 KL TL TK KL LK TK KL Substitutin eq. (22) in (8), we derive ( K 2[ TLLK ( KL TK ) TL TK KL] L2 KL TL TK KLLK TK KL X '/ ' [( ) ]) ( K 2TL L2 KL) LK [ K 2( TK ) L2( TK )] K2[ ( KL TK )] TL L2( KL TL) TK KL. (23) Recall eq. (5), that is, ih ( h / i) hi, i, h T, K, L. Usin this, eliminate KL, TL, TK, and use LK, LT, KT : L ( K 2 TL L2 KL) LK [ K 2( TK ) L2( TK )] LK( ) K X '/ '. L K K2[ ( KL TK )] LT( ) L2( KL TL) KT( ) T T (24) For example, from eq. (2), we can easily show that KL TL K L2 K 2 L T L2 T 2 L ( ) ( ) ( ) ( ) (25) L K 2 T 2 L2 K T L L2 L2 L L L2 E ( ), ABE,, (, 2, L L2). Substitutin eq. where recall eq. (22), that is, T T 2 K K (25) in (24), we have for the term of KT : K K L2( KL TL ) KT( ) L2( E) KT( ). T T (26) From eqs (6) and (7), we derive the equation as follows. X X wt '/ ' */ *. (27) L.H.S. is the enery tariff commodity output relationship in Thompson (204). R.H.S.is the enery price commodity output relationship in the 3 x 2 model with no enery tariff shown in this article. 28

29 This implies that eq. (24) must be equivalent to (58). Hence, eq. (26) must be equivalent to the term of U( KT) in eq. (58), that is, EL2U ( EL2KT). (28) However, this doesn t hold. Similarly, we can show that other terms in eq. (24) also include errors. Apparently, Thompson s equation includes an error. On the other hand, Thompson (204, eq. ()) showed the equation, which expresses the effects of prices on enery imports ( E ): E'/ p' ( K2L22) /, (29) where ( ) ( ), ( ) ( ) LK. (30) KL EK EL EL EK KL 2 KL EL EK EK EL Replacin E and ih with T and ih, we derive ( ) ( ), ( ) ( ). (3) KL TK TL TL TK KL 2 KL TL TK TK TL LK Substitutin eq. (3) in (29), we derive: ( K 2[ ( KL TK ) TL ( TL TK ) KL] E'/ p' L2[ ( KL TL) TK ( TK TL) LK] ) K2( TL TK ) KL L2( TK TL) LK. K2( KL TK ) TL L2( KL T L) TK (32) Recall eq. (5), that is, ih ( h / i) hi, i, h T, K, L. Usin this, eliminate KL, TL, TK, and use LK, LT, KT : E L K2( TL TK ) LK( ) L2( TK TL) LK K '/ p'. L K K2( K L TK ) LT( ) L2( KL TL ) KT( ) T T (33) For example, from eq. (2), we can show that 29

30 ( ) KL TL K L2 K 2 L T L2 T 2 L K K 2 T T 2 K K T T L L2 L L2 L L L L ( ) ( ). K L T L (34) Substitutin eq. (34) in (33), we have for the term of K L2[( K L ) ( T L )] KT( ). T KT : (35) If we use our expression, we derive the equation as follows. E '/ p' VT * p*. (36) This implies that eq. (33) must be equivalent to (86). Hence, eq. (35) must be equivalent to the term of U( KT) in eq. (86), that is, EL2U ( EL2KT ). (37) T T However, this doesn t hold. Similarly, we can show that other terms in eq. (33) also include errors. Apparently, Thompson s equation includes an error. References: Bandyopadhyay, Taradas and Bharati Bandyopadhyay, 989, Economics of the import of factors of production: Comparative advantae and commercial policy, European Journal of Political Economy, 5, Batra, R.N. and Casas, F.R. (976), A synthesis of the Heckscher-Ohlin and the neoclassical models of international trade, Journal of International Economics, 6, Brecher, R. and Findlay, R. (983), Tariffs, Forein Capital and National Welfare with Sector-Specific Factors, Journal of International Economics, 4, Chipman, J. 97, International trade with Capital Mobility: A Substitution Theorem, in J. N. Bhawati, R. W. Jones, R. A. Mundell, J. Vanek, eds., Trade, balance of payments and rowth: papers in international economics in honor of Charles P. Kindleberer, Amsterdam: North- Holland Pub. Co. pp Dei, Fumio, 985, Welfare ains from capital inflows under import quotas, Economics Letters, 8, Ethier, Bill, and Svensson, Lars, 986. The theorems of international trade and factor mobility. 30

31 Journal of International Economics, 20, Feruson, David, 978. International capital mobility and comparative advantae: the two-country, two-factor case. Journal of International Economics, 8, Fullerton, Don, and Heutel, Garth 2007, The General Equilibrium Incidence of Environmental Taxes. Journal of Public Economics 9, Heutel, Garth and Fullerton, Don, (200), "Analytical General Equilibrium Effects of Enery Policy on Output and Factor Prices". Economics Faculty Publications. Paper 3. Ishikawa, J. and K. Kiyono (2006) Greenhouse-Gas Emission Controls in an Open Economy, International Economic Review, Vol. 47, pp Ishikawa, J., K. Kiyono, and M. Yomoida (202) Is Emission Tradin Beneficial?, The Japanese Economic Review, Vol. 63, No. 2, p Jones, R., 967, International Capital Movements and the Theory of Tariffs and Trade, The Quarterly Journal of Economics, 8, -38. Jones, R.W. and Easton, S.T. (983) Factor intensities and factor substitution in eneral equilibrium, Journal of International Economics, 5, Jones, R. and Ruffin, R., 975, Trade patterns with Capital Mobility, in M. Parkin and A.R. Nobay, eds., Current economic problems: the proceedins of the [annual meetin of the] Association of University Teachers of Economics, Manchester 974, Cambride: Cambride University Press, pp Kemp, Murray C., (966), The Gain from International Trade and Investment: A Neo-Heckscher- Ohlin Approach, The American Economic Review, Vol. 56, No. 4, Part, pp Mundell, Robert, 957. International trade and factor mobility. American Economic Review, 47, Nakada, Y. (206a), Factor endowment commodity output relationships in a three-factor two-ood eneral equilibrium trade model, Available at. Nakada, Y. (206b), Factor endowment commodity output relationships in a three-factor two-ood eneral equilibrium trade model, Further analysis Available at. Nakada, Y. (206c), Derivin the factor endowment commodity output relationship for Thailand ( ) usin a three-factor two-ood eneral equilibrium trade model, Available at 3

32 Oino, Kazunori, 990, The Equivalence of Tariffs and Quotas under International Capital Movements, The Economic Studies Quarterly, Vol. 4, No. 2, (in Japanese). Ruffin, R.J. (98), Trade and factor movements with three factors and two oods, Economics Letters, 7, Samuelson, P.A. (953), Foundations of economic analysis, Cambride: Harvard University Press. Samuelson, P.A. (983), Foundations of economic analysis, enlared edition, Cambride: Harvard University Press [Keizai Bunseki no Kiso, Zoho ban. Tokyo: Keiso-shobo (translated by Sato Ryuzo in 986 in Japanese)]. Sato, R. and Koizumi, T. (973), On the Elasticities of Substitution and Complementarity, Oxford Economic Papers, 25, Srinivasan, T.N., 983. International factor movements, commodity trade and commercial policy in a specific factors model. Journal of International Economics, 2, Styan, G.P.H. (973), Hadamard products and multivariate statistical analysis, Linear Alebra and its Applications, 6, Svensson, Lars, 984. Factor trade and oods trade. Journal of International Economics, 6, Takeda, Shiro, (2005) "The effect of differentiated emission taxes: does an emission tax favor industry?" Economics Bulletin, Vol. 7, No. 3 pp. 3. Thompson, H. (983), Trade and international factor mobility, Atlantic Economic Journal,, Thompson, Henry, 985, International Capital Mobility in a Specific Factor Model, Atlantic economic journal, Vol.3, No.2, Thompson, H. (994), Do oil tariffs lower waes? Open Economies Review, 5, Thompson, H, (2000), Enery taxes and waes in eneral equilibrium. OPEC Review, 24, Thompson, H. (204), Enery tariffs in a small open economy, Enery Economics, 44,

33 U'-axis quad. II line T2 quad. I R T2 P3 line T P2 R T O S'-axis P Q M quad. III quad. IV Fi. Illustration of the EWS-ratio vector boundary and line-tj (border line for a sin pattern of X j */w T * to chane) Note: S' = S/T = LK / LT, U' = U/T = KT / LT 33

34 U'-axis quad. II line 2 quad. I R T2 Pb Pa R T O S'-axis Ma quad. III quad. IV Fi. 2 Illustration of the EWS-ratio vector boundary and line 2 (border line for a sin pattern of X */p 2 * to chane) Note: S' = S/T = LK / LT, U' = U/T = KT / LT 34