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Asymmetric adjustment costs in simple general equilibrium models
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European Economic Review 48 (2004) 63 – 73
www.elsevier.com/locate/econbase
Asymmetric adjustment costs in simple general equilibrium models Avik Chakrabarti∗ Department of Economics, University of Wisconsin, 816 Bolton Hall, Milwaukee, WI 53201, USA Accepted 13 June 2002
Abstract This paper demonstrates that the introduction of asymmetric adjustment costs in a simple general equilibrium framework establishes a meaningful link between factor price determination and output determination, breaking the analytically convenient dichotomy of the Heckscher– Ohlin–Samuelson model. The possibility of trade between seemingly similar countries that di0er in their adjustment technologies is visited. c 2002 Elsevier B.V. All rights reserved. JEL classi'cation: F1; D5 Keywords: General equilibrium; International trade; Heckscher–Ohlin–Samuelson theory; Adjustment costs; Factor price determination
1. Introduction The role of adjustment costs in a general equilibrium model is widely recognized. 1 A number of econometric studies have also provided estimates of adjustment costs. 2 However, there is no compelling reason to believe that such adjustment costs will be ∗
Tel.: +1-414-229-4680; fax: +1-414-229-3860. E-mail address: [email protected] (A. Chakrabarti). 1 For an early treatment of adjustment costs of investment see Lucas (1967) where he suggested that this cost behavior can be thought of as a sum of purchase costs (with perfect or imperfect factor markets) and installation costs. Mussa (1978) was the Brst to recognize the role of economic resources in the movement of capital from one sector to another. He demonstrated how a balance between expectations of future returns and costs of capital movement determine the eCciency of the adjustment process. See Mulligan and Sala-i-Martin (1993) for an insightful discussion on this issue. 2 See Barnett and Sakellaris (1999), Kolstad and Lee (1993), Wolfson (1993), Zanias (1991), Bernstein (1988), Lichtenberg (1988), Mohnen et al. (1986), and Pindyck and Rotemberg (1983) for representative estimates of adjustment costs. c 2002 Elsevier B.V. All rights reserved. 0014-2921/$ - see front matter PII: S 0 0 1 4 - 2 9 2 1 ( 0 2 ) 0 0 2 5 8 - 1
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A. Chakrabarti / European Economic Review 48 (2004) 63 – 73
uniform across sectors. 3 This paper demonstrates that the introduction of asymmetric adjustment costs in an otherwise Heckscher–Ohlin–Samuelson (H–O–S) model establishes a meaningful link between factor price determination and output determination, breaking the analytically convenient dichotomy. In consequence, the possibility of the emergence of a trading equilibrium between two economies that di0er in their adjustment technologies but are otherwise identical is demonstrated. The rest of the paper is organized as follows. Section 2 presents a diagrammatic analysis of the e0ects of asymmetric adjustment costs. It provides a useful taxonomy for the formal model presented in Section 3. Section 4 concludes. 2. A diagrammatic analysis Visualize a small open economy that uses two factors (in Bxed supply), capital (K) and labor (L) to produce two goods, X (labor intensive) and Y (capital intensive), with constant returns to scale technology. Let x and y denote the quantities of X and Y , respectively. Let w denote the purchase price of labor and r the purchase price of capital, Kj and Lj denote the capital and labor employed in sector j, Pj denote the world price of good j, where j = X , Y . Labor is freely mobile between sectors. Capital is a quasi-Bxed input in the sense that adding new capital, removing existing capital and/or moving capital from one sector to another entails an adjustment cost. Moving capital from one sector to another is viewed as a two-step process: (1) Removal: Capital is removed from one sector, the cost of which is borne exclusively by that sector and (2) addition: The released capital is transformed and installed in the other sector, the cost of which is borne exclusively by the recipient sector. To focus on the asymmetry in adjustment costs between sectors I ignore any di0erence in adjustment cost (a) between addition of capital to a sector and removal of capital from the same sector and (b) between addition of new capital to a sector and addition of capital to the same sector released by the other. For simplicity, I also assume that capital is the only resource used up in the installation and implementation of capital. Consider an expansion in the endowment of capital. Let the marginal cost of adjustment be higher in the capital-intensive sector relative to the labor-intensive sector. Let aj denote a constant marginal cost of adjustment in sector j. In transition, the factor cost ratios faced by the producers of X and Y goods will then bear the following relationship: (w=r) ¿ [w=(r + aX )]X ¿ [w=(r + aY )]Y : 3
(1)
Chakrabarti (1999) presents evidence from U.S. industries that indicate systematic Di0erences in costs of adjustment associated with the introduction of new capital across sectors di0ering in factor-intensities. A panel of annual observations on 457 industries by 4-digit SIC in the U.S. economy over the period 1958–1994 indicates that, on an average, the estimated cost of adjustment is signiBcantly higher in the capital-intensive sectors than it is in the labor-intensive sectors. A panel of annual observations on 521 Brms from the U.S. economy over the period 1984 –1992 leads to a similar conclusion: the estimated cost of adjustment is signiBcantly higher in the relatively capital-intensive group of Brms than it is in the labor-intensive group. Benoit and Serena (1992) also report signiBcant di0erences in the speed and the costs of adjustment in Canadian industries.
A. Chakrabarti / European Economic Review 48 (2004) 63 – 73
65
Fig. 1. The Jones–Rybczynski e0ect.
Fig. 2. AAC and excess supply of capital.
E0ects the expansion of capital will have on the composition of output are shown in Figs. 1–4. 4 The solid lines indicate unit-value isoquants, unit isocost lines, and allocation of factor endowment (E) for capital and labor before the expansion of capital. The broken lines represent the situation after the expansion of capital forming the new endowment (E ). Fig. 1 captures the familiar Jones–Rybczynski (JR) e0ect in the absence of any adjustment cost. An increase in capital moves endowment from E to E . Relative factor prices and capital intensities remain unchanged. The new output composition is represented by the vectors OC and OD replacing OA and OB for goods Y and X , respectively: Y expands and X contracts. 4
For simplicity of diagrammatic exposition I have assumed, in this section, that production and adjustment are integrated within each sector. To avoid any confusion it is important that the reader notes the distinction between e0ective endowment e (net of the capital used up exclusively for transformation of capital, measured by the broken line E e ) and actual endowment E in Figs. 3 and 4. In the following section, the formal analysis recognizes production and adjustment as two separate activities.
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A. Chakrabarti / European Economic Review 48 (2004) 63 – 73
Fig. 3. Labor-intensive sector expands; capital-intensive sector contracts.
Figs. 2–4 demonstrate the e0ects that the same expansion of capital can have in the presence of asymmetric adjustment costs (AAC). In Fig. 2, the two broken lines indicate the di0erent factor cost ratios (as in inequality (2)) to be faced by each sector in transition. Clearly, neither sector has any incentive to expand or contract. This results in an excess supply of capital (measured by the broken line EE ). This excess supply pressure will lower the purchase price of capital (r). When r falls enough to o0set the adjustment cost of capital in sector X (i.e. when the broken iso-cost line for sector X swings to the right of the solid iso-cost line) Brms in sector X will Bnd it proBtable to expand production. 5 Wage (w) will rise in the face of expansion of the labor-intensive sector resulting in a rise in the purchase price of labor. At a new factor purchase price ratio (w =r ) and the corresponding factor cost ratios (w =(r + aX ) for X and w =(r + aY ) for Y ) full employment will be achieved. Capital intensities will rise in both sectors. Figs. 3 and 4 depict three possible e0ects on output composition. The new output composition is represented by the rays OF and OG for goods Y and X , respectively. A thick isoquant is drawn to indicate the initial level of output in each sector. The e0ective endowment (net of the capital used up exclusively for installation and implementation of capital, measured by the broken line E e ) is represented by e . In Fig. 3, X expands and Y contracts. In Fig. 4, X and Y both expand. The Bgures reveal the possibility 5 Note that as r continues to fall, sector X will have the incentive to expand Brst when the adjustment cost is higher in sector Y .
A. Chakrabarti / European Economic Review 48 (2004) 63 – 73
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Fig. 4. Both (labor-intensive and capital-intensive) sectors expand.
that if capital becomes cheaper, capital–labor ratios in both sectors can rise, and this can favor output expansion of the labor-intensive good at the expense of the other. The intuition is as follows. In the H–O–S model, absent any asymmetry in costs of adjustment, the dichotomy allows reallocation of resources between industries without a0ecting factor prices, following any change in factor endowments within the cone of diversiBcation. 6 If adjustment costs are asymmetric an expansion in the economy’s capital stock generates two e0ects, namely, an endowment e7ect and a substitution e7ect. At unchanged relative factor prices the endowment e0ect would induce an expansion in the capital-intensive sector and a contraction in the labor-intensive sector generating the familiar JR e0ect. When there are costs of adjustment neither sector would have an incentive to expand or contract. This results in an excess supply of capital that lowers the purchase price of capital relative to that of labor. The decline 6
See Chakrabarti (1998) for a generalized factor price equalization (FPE) theorem which identiBes a condition, stated in terms of the allocation of factor endowments across countries relative to the demand for and the factor intensities of goods, that is necessary and su8cient for FPE in a world with arbitrary number of countries, goods and factors. Deardor0 (1994) suggested a “lens” condition for FPE that he showed is necessary and conjectured is suCcient in a world with arbitrary number of countries, goods and factors. Chakrabarti (1994) proved that the lens condition is necessary and suCcient for FPE i7 the world has at least as many goods as factors.
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in the relative purchase price of capital induces each sector to substitute away from labor. This substitution e0ect operates against the endowment e0ect. The change in the output of each sector depends on the relative strength of the endowment e0ect and the substitution e0ect. 3. A formal analysis The analysis presented in this section builds on the structure of simple general equilibrium models developed by Jones (1965). 7 A variable with ∧ denotes the percentage change in that variable. Let aij represent the amount of factor i used to produce one unit of good j where i = K, L; LQ and KQ represent the supply of capital and labor. In addition to the two sectors producing tradable goods (X and Y ) there is a competitive sector (T ) which exclusively provides a non-tradable service that transforms capital (new or released) to Bt its use in each of the tradable goods sectors: KAj represents the amount of capital used by T exclusively for adjustment in the output of sector j; Paj is the unit cost of adjustment for sector j (i.e. the price paid by sector j to T in transforming capital to support a unit change in the output of sector j). The adjustment technology is speciBed as KAj = j dj; where
j =
(2)
+j
for dj ¿ 0;
j
for dj 6 0;
−
and j ¿ 0 are constants. All other notations used in this section are consistent with those used in Section 2. In a competitive equilibrium the zero-proBt condition requires: wLX + rKX + PaX dx = PX x;
(3a)
wLY + rKY + PaY dy = PY y;
(3b)
PaX PaY =r= : Y X
(3c)
Full employment of resources ensures: Q LX + LY = L;
(4a)
Q KX + KY + KAX + KAY = K:
(4b)
7 See also Chakrabarti (2001), Jones and Marjit (1985), Marjit (1987, 1990, 1993), Marjit and Beladi (1990), and Jones et al. (1999).
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Conditions (3a)–(4b) reduce to the following equations of change: ˆ LX wˆ + KX rˆ = Pˆ X − AX x;
(5a)
ˆ LY wˆ + KY rˆ = Pˆ Y − AY y;
(5b)
LX xˆ + LY yˆ = Lˆ + L (wˆ − r); ˆ
(5c)
ˆ ˜KX xˆ + ˜KY yˆ = Kˆ − K (wˆ − r);
(5d)
where Lj = (waLj =Pj ) and Kj = (raKj =Pj ), ij being the distributive share of factor i in industry j; Lj = Lj =L and Kj = Kj =K are fractions of the economy’s labor and K capital employed in the production of good j; Aj = ((j r)=Pj ); Aj = ((j j)=K); and K ˜ Kj = (Kj + Aj ). Let Kˆ = Lˆ = Pˆ X = Pˆ Y = 0 in the initial equilibrium. Consider the e0ect that a subsequent expansion in capital stock (i.e. Kˆ ¿ 0) has on the composition of output. Let commodity prices and labor supply remain unchanged, i.e. Lˆ = Pˆ X = Pˆ Y = 0. Under these simpliBcations, the relative change in the factor prices is given by Y yˆ − X xˆ (wˆ − r) ˆ = ; (6) || where || = [(KY − KX ) − (AX KY − AY KX )]; X = (KY + LY )AX and Y = (KX + LX )AY . It may be noted that absent any cost of adjustment (i.e. j = 0) relative factor rewards will not change in response to the rise in capital stock. Let L = (LX KX X + LY KY Y ) ¿ 0
(7a)
K = (KX LX X + KY LY Y ) ¿ 0;
(7b)
and where j is the elasticity of substitution between labor and capital in industry j. From (5c), (5d) and (6), the change in the output of each sector can be expressed as Y ˆ xˆ = K; (8a) || −X ˆ yˆ = K; (8b) || where X = (LX + (L X )=||); Y = (LY − (L Y )=||); X = (˜KX − (K X )=||); Y =(˜KY +(K Y )=||); and ||=(X Y −Y X ). (Lj =||)Kˆ captures the endowment e0ect and (L j =||)Kˆ the substitution e0ect. The endowment e0ect isolates the direct e0ect that an expansion in capital (net of the capital used in adjustment) has on the output of each sector at any given factor price ratio. The substitution e0ect arises from any change in the relative factor rewards following the excess supply of capital at the initial factor price ratio.
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Absent any cost of adjustment (i.e. j = 0) Eqs. (8a) and (8b) yield the familiar JR ˆ e0ect: (xˆ − y) ˆ = −K=|| where || = (KY − KX ) ∈ (0; 1). The capital-intensive sector expands proportionately more than the rise in the capital stock (Jones’ magniBcation e0ect) and the labor-intensive sector contracts. From (8a) and (8b), Y xˆ =− : (9) yˆ X Eq. (9) characterizes a generalized (non-linear) JR path of expansion when there are costs of adjustment. If X and Y have the same sign then one of the sectors expands at the cost of another. A condition suCcient to induce an expansion in both sectors is KY X KX ¡ : and ¡ (1 − AX ) LX L (1 − AY ) If
|X | || ¿ ; LX L
Y || ¡ LY L
and
X ¿ aKX ;
then X will contract and Y will expand. If, on the other hand, KY =(1 − AY ) ¿ KX = (1 − AX ) and Y ¿ aKY , then X will expand and Y will contract. There are interesting implications of these results for international trade. For instance, let two small economies (foreign and home) be characterized by identical endowments and preferences, share identical production technologies, and be both initially in autarky. Let the free trade price (pf ) be equal to the initial autarkic price. Let an asterisk represent the foreign country and no-asterisk the home country. Consider the possibility that adjustment technologies in the two countries exhibit the following properties: KY |∗X | | ∗ | ∗Y | ∗ | KX ¿ ; ¿ ; Y ¿ aKY ; ¿ ∗ ∗ (1 − AY ) (1 − AX ) LX ∗L LY ∗L and
X ∗ ¿ a∗KX :
Let both these economies experience an identical expansion in capital. The endowment e0ect for both these economies will be identical. The substitution e0ect for the foreign economy will reinforce the endowment e0ect and it will experience the JR e0ect: The capital-intensive sector will expand and the labor-intensive sector will contract. For the home country the substitution e0ect will operate against the endowment e0ect. If the home substitution e0ect is strong enough to more than o0set the endowment e0ect then the JR e0ect will be reversed: The labor-intensive sector will expand and the capital-intensive sector will contract. This can result in a trading equilibrium where the home country exports its labor-intensive good to the foreign country and the foreign country exports its capital-intensive good to the home country. Such a trading equilibrium between two economies that di0er in their adjustment technologies but are otherwise identical is depicted in Fig. 5, drawn under the simplifying assumption of homothetic preferences. 8 Initially production and consumption is at A for both 8 This argument provides a plausible explanation for the large volume of North–North trade relative to North–South trade.
A. Chakrabarti / European Economic Review 48 (2004) 63 – 73
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Fig. 5. Balanced trade between two identical countries but for an asymmetry in adjustment costs: Home country exports the labor-intensive good and foreign country exports the capital-intensive good.
countries. The production possibility frontier shifts from TT to T T (net of the capital used up in adjustment) after the expansion in capital. Production in the home country moves to P and that in the foreign country moves to P∗ . 9 At pf consumption in the home country moves to C and that in the foreign country moves to C∗ resulting in a trading equilibrium where the home country exports X (labor intensive) and the foreign country exports Y (capital intensive). 10 4. Conclusion This paper demonstrates that the introduction of asymmetric adjustment costs in a simple general equilibrium framework establishes a meaningful link between factor price determination and output determination, breaking the analytically convenient dichotomy of the H–O–S model. It is shown that in a two-sector model an expansion in the endowment of a factor (capital) can lead to an expansion, contraction or no change in the output of either sector depending on the adjustment technology. The possibility of a trading equilibrium between two economies that di0er in their adjustment technologies but are otherwise identical is demonstrated. Finally, innumerous attempts, in a 9 The slope at any point on T T reTects the ratio (MC =MC ) of marginal costs (MC) of production X Y and does not incorporate the costs of adjustment. At any given point on T T the true MC ratio (mcX =mcY ), after incorporating costs of adjustment, is higher in the foreign country than in the home country. The output mix is chosen by each country in a way that mcX =mcY is equal the free trade price ratio (pf ). 10 This argument may have implications for the Leontie0 paradox as well. For instance, in 1947, at a time when the U.S. was considered to be the most capital-rich country, the capital-labor ratio embodied in U.S. imports exceeded the ratio embodied in exports by 60% (Leontie0, 1954). This apparently “paradoxical” pattern may very well have been due to signiBcant di0erences in adjustment costs across sectors. It may also be argued that any asymmetry in costs of adjustment is a relatively short-term phenomenon and a convergence in adjustment costs over the long run can explain the fact that the Leontie0 paradox has systematically been observed to have gradually weakened, though not disappeared, in later data sets.
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vast pool of empirical literature, are continuously made to test (weak) implications of the H–O–S theory. Most studies report that the implications are spectacularly at odds with the description of international data. 11 By bringing out the implications of asymmetric adjustment costs this paper points to the importance of allowing sector-speciBc adjustment costs in tests of the implications of the H–O–S theory. References Barnett, S.A., Sakellaris, P., 1999. A new look at Brm market value, investment, and adjustment costs. The Review of Economics and Statistics 81, 250–260. Benoit, C., Serena, Ng., 1992. Adjustment costs and factor demands in Canadian manufacturing industries. Applied Economics 24, 845–857. Bernstein, J.I., 1988. Dynamic factor demands and adjustment costs: An analysis of Bell Canada’s technology. Information Economics and Policy 3, 5–24. Chakrabarti, A., 1994. Proof of suCciency of the lens condition for factor price equalization. Mimeo., University of Michigan, Ann Arbor, MI. Chakrabarti, A., 1998. A generalized factor price equalization theorem. Mimeo., University of Wisconsin, Milwaukee, WI. Chakrabarti, A., 1999. Estimation of internal adjustment costs: Capital-intensive and labor-intensive industries. Mimeo., University of Wisconsin, Milwaukee, WI. Chakrabarti, A., 2001. Rybczynski theorem and asymmetric adjustment costs. Keio Economic Studies 38, 33–41. Davis, D.R., Weinstein, D.E., Bradford, S.C., Shimpo, K., 1997. Using international and Japanese regional data to determine when the factor abundance theory of trade works. American Economic Review 87, 421– 446. Deardor0, A., 1994. The possibility of factor price equalization revisited. Journal of International Economics 47, 167–175. Helpman, E., 1999. The structure of foreign trade. Journal of Economic Perspectives 13, 121–144. Jones, R.W., 1965. The structure of simple general equilibrium models. Journal of Political Economy 73, 557–572. Jones, R.W., Marjit, S., 1985. A simple production model with Stolper–Samuelson properties. International Economic Review 26, 565–567. Jones, R.W., Marjit, S., Beladi, H., 1999. Three faces of factor intensities. Journal of International Economics 48, 413–420. Kolstad, C.D., Lee, L., 1993. The speciBcation of dynamics in cost function and factor demand estimation. The Review of Economics and Statistics 75, 721–726. Leamer, E.E., Levinsohn, J.A., 1995. International trade theory: The evidence. In: Grossman, G.M., Rogo0, K. (Eds.), Handbook of International Economics, Vol. 3. North-Holland, Amsterdam, pp. 1339 –1394. Leontie0, W.W., 1954. Domestic production and foreign trade: The American capital position re-examined. Proceedings of the American Philosophical Society 97, 332–349. Lichtenberg, F.R., 1988. Estimation of the internal adjustment costs model using longitudinal establishment data. The Review of Economics and Statistics 70, 421–430. Lucas, R., 1967. Adjustment costs and the theory of supply. Journal of Political Economy 75, 321–334. Marjit, S., 1987. Trade in intermediates and the colonial pattern of trade. Economica 54, 173–184. Marjit, S., 1990. A simple production model in trade and its applications. Economics Letters 32, 257–260. Marjit, S., 1993. Uniform tari0s in general equilibrium: A simple model. Zeitschrift f uV r Nationalokonomie 57, 189–196. 11 See TreTer (1995), Davis et al. (1997) for recent accounts of ways in which international data deviate from the theoretical predictions of the Heckscher–Ohlin model and a plausible reconciliation in terms of cross-country productivity di0erences and home bias in demand. See also Leamer and Levinsohn (1995) and Helpman (1999) for comprehensive reviews of the relevant literature.
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Marjit, S., Beladi, H., 1990. Factor underutilization and the theory of international trade. Keio Economic Studies 27, 43–51. Mohnen, P.M., Nadiri, I., Prucha, I.R., 1986. R&D, production structure, and productivity growth in the U.S., Japanese and German manufacturing sectors. European Economic Review 30, 749–771. Mulligan, C.B., Sala-i-Martin, X., 1993. Transitional dynamics in two-sector models of endogenous growth. The Quarterly Journal of Economics 108, 739–773. Mussa, M., 1978. Dynamic adjustment in the Heckscher–Ohlin–Samuelson model. Journal of Political Economy 86, 775–791. Pindyck, R., Rotemberg, J., 1983. Dynamic factor demands under rational expectations. Scandinavian Journal of Economics 85, 223–238. TreTer, D., 1995. The case of missing trade and other mysteries. American Economic Review 85, 1029–1046. Wolfson, P., 1993. Compositional change, aggregation, and dynamic factor demand: estimates on a panel of manufacturing Brms. Journal of Applied Econometrics 8, 129–148. Zanias, G.P., 1991. Adjustment costs, production functions and capital–labour substitution in Greek manufacturing: 1958–80. Applied Economics 23, 49–55.
European Economic Review 48 (2004) 63 – 73
www.elsevier.com/locate/econbase
Asymmetric adjustment costs in simple general equilibrium models Avik Chakrabarti∗ Department of Economics, University of Wisconsin, 816 Bolton Hall, Milwaukee, WI 53201, USA Accepted 13 June 2002
Abstract This paper demonstrates that the introduction of asymmetric adjustment costs in a simple general equilibrium framework establishes a meaningful link between factor price determination and output determination, breaking the analytically convenient dichotomy of the Heckscher– Ohlin–Samuelson model. The possibility of trade between seemingly similar countries that di0er in their adjustment technologies is visited. c 2002 Elsevier B.V. All rights reserved. JEL classi'cation: F1; D5 Keywords: General equilibrium; International trade; Heckscher–Ohlin–Samuelson theory; Adjustment costs; Factor price determination
1. Introduction The role of adjustment costs in a general equilibrium model is widely recognized. 1 A number of econometric studies have also provided estimates of adjustment costs. 2 However, there is no compelling reason to believe that such adjustment costs will be ∗
Tel.: +1-414-229-4680; fax: +1-414-229-3860. E-mail address: [email protected] (A. Chakrabarti). 1 For an early treatment of adjustment costs of investment see Lucas (1967) where he suggested that this cost behavior can be thought of as a sum of purchase costs (with perfect or imperfect factor markets) and installation costs. Mussa (1978) was the Brst to recognize the role of economic resources in the movement of capital from one sector to another. He demonstrated how a balance between expectations of future returns and costs of capital movement determine the eCciency of the adjustment process. See Mulligan and Sala-i-Martin (1993) for an insightful discussion on this issue. 2 See Barnett and Sakellaris (1999), Kolstad and Lee (1993), Wolfson (1993), Zanias (1991), Bernstein (1988), Lichtenberg (1988), Mohnen et al. (1986), and Pindyck and Rotemberg (1983) for representative estimates of adjustment costs. c 2002 Elsevier B.V. All rights reserved. 0014-2921/$ - see front matter PII: S 0 0 1 4 - 2 9 2 1 ( 0 2 ) 0 0 2 5 8 - 1
64
A. Chakrabarti / European Economic Review 48 (2004) 63 – 73
uniform across sectors. 3 This paper demonstrates that the introduction of asymmetric adjustment costs in an otherwise Heckscher–Ohlin–Samuelson (H–O–S) model establishes a meaningful link between factor price determination and output determination, breaking the analytically convenient dichotomy. In consequence, the possibility of the emergence of a trading equilibrium between two economies that di0er in their adjustment technologies but are otherwise identical is demonstrated. The rest of the paper is organized as follows. Section 2 presents a diagrammatic analysis of the e0ects of asymmetric adjustment costs. It provides a useful taxonomy for the formal model presented in Section 3. Section 4 concludes. 2. A diagrammatic analysis Visualize a small open economy that uses two factors (in Bxed supply), capital (K) and labor (L) to produce two goods, X (labor intensive) and Y (capital intensive), with constant returns to scale technology. Let x and y denote the quantities of X and Y , respectively. Let w denote the purchase price of labor and r the purchase price of capital, Kj and Lj denote the capital and labor employed in sector j, Pj denote the world price of good j, where j = X , Y . Labor is freely mobile between sectors. Capital is a quasi-Bxed input in the sense that adding new capital, removing existing capital and/or moving capital from one sector to another entails an adjustment cost. Moving capital from one sector to another is viewed as a two-step process: (1) Removal: Capital is removed from one sector, the cost of which is borne exclusively by that sector and (2) addition: The released capital is transformed and installed in the other sector, the cost of which is borne exclusively by the recipient sector. To focus on the asymmetry in adjustment costs between sectors I ignore any di0erence in adjustment cost (a) between addition of capital to a sector and removal of capital from the same sector and (b) between addition of new capital to a sector and addition of capital to the same sector released by the other. For simplicity, I also assume that capital is the only resource used up in the installation and implementation of capital. Consider an expansion in the endowment of capital. Let the marginal cost of adjustment be higher in the capital-intensive sector relative to the labor-intensive sector. Let aj denote a constant marginal cost of adjustment in sector j. In transition, the factor cost ratios faced by the producers of X and Y goods will then bear the following relationship: (w=r) ¿ [w=(r + aX )]X ¿ [w=(r + aY )]Y : 3
(1)
Chakrabarti (1999) presents evidence from U.S. industries that indicate systematic Di0erences in costs of adjustment associated with the introduction of new capital across sectors di0ering in factor-intensities. A panel of annual observations on 457 industries by 4-digit SIC in the U.S. economy over the period 1958–1994 indicates that, on an average, the estimated cost of adjustment is signiBcantly higher in the capital-intensive sectors than it is in the labor-intensive sectors. A panel of annual observations on 521 Brms from the U.S. economy over the period 1984 –1992 leads to a similar conclusion: the estimated cost of adjustment is signiBcantly higher in the relatively capital-intensive group of Brms than it is in the labor-intensive group. Benoit and Serena (1992) also report signiBcant di0erences in the speed and the costs of adjustment in Canadian industries.
A. Chakrabarti / European Economic Review 48 (2004) 63 – 73
65
Fig. 1. The Jones–Rybczynski e0ect.
Fig. 2. AAC and excess supply of capital.
E0ects the expansion of capital will have on the composition of output are shown in Figs. 1–4. 4 The solid lines indicate unit-value isoquants, unit isocost lines, and allocation of factor endowment (E) for capital and labor before the expansion of capital. The broken lines represent the situation after the expansion of capital forming the new endowment (E ). Fig. 1 captures the familiar Jones–Rybczynski (JR) e0ect in the absence of any adjustment cost. An increase in capital moves endowment from E to E . Relative factor prices and capital intensities remain unchanged. The new output composition is represented by the vectors OC and OD replacing OA and OB for goods Y and X , respectively: Y expands and X contracts. 4
For simplicity of diagrammatic exposition I have assumed, in this section, that production and adjustment are integrated within each sector. To avoid any confusion it is important that the reader notes the distinction between e0ective endowment e (net of the capital used up exclusively for transformation of capital, measured by the broken line E e ) and actual endowment E in Figs. 3 and 4. In the following section, the formal analysis recognizes production and adjustment as two separate activities.
66
A. Chakrabarti / European Economic Review 48 (2004) 63 – 73
Fig. 3. Labor-intensive sector expands; capital-intensive sector contracts.
Figs. 2–4 demonstrate the e0ects that the same expansion of capital can have in the presence of asymmetric adjustment costs (AAC). In Fig. 2, the two broken lines indicate the di0erent factor cost ratios (as in inequality (2)) to be faced by each sector in transition. Clearly, neither sector has any incentive to expand or contract. This results in an excess supply of capital (measured by the broken line EE ). This excess supply pressure will lower the purchase price of capital (r). When r falls enough to o0set the adjustment cost of capital in sector X (i.e. when the broken iso-cost line for sector X swings to the right of the solid iso-cost line) Brms in sector X will Bnd it proBtable to expand production. 5 Wage (w) will rise in the face of expansion of the labor-intensive sector resulting in a rise in the purchase price of labor. At a new factor purchase price ratio (w =r ) and the corresponding factor cost ratios (w =(r + aX ) for X and w =(r + aY ) for Y ) full employment will be achieved. Capital intensities will rise in both sectors. Figs. 3 and 4 depict three possible e0ects on output composition. The new output composition is represented by the rays OF and OG for goods Y and X , respectively. A thick isoquant is drawn to indicate the initial level of output in each sector. The e0ective endowment (net of the capital used up exclusively for installation and implementation of capital, measured by the broken line E e ) is represented by e . In Fig. 3, X expands and Y contracts. In Fig. 4, X and Y both expand. The Bgures reveal the possibility 5 Note that as r continues to fall, sector X will have the incentive to expand Brst when the adjustment cost is higher in sector Y .
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Fig. 4. Both (labor-intensive and capital-intensive) sectors expand.
that if capital becomes cheaper, capital–labor ratios in both sectors can rise, and this can favor output expansion of the labor-intensive good at the expense of the other. The intuition is as follows. In the H–O–S model, absent any asymmetry in costs of adjustment, the dichotomy allows reallocation of resources between industries without a0ecting factor prices, following any change in factor endowments within the cone of diversiBcation. 6 If adjustment costs are asymmetric an expansion in the economy’s capital stock generates two e0ects, namely, an endowment e7ect and a substitution e7ect. At unchanged relative factor prices the endowment e0ect would induce an expansion in the capital-intensive sector and a contraction in the labor-intensive sector generating the familiar JR e0ect. When there are costs of adjustment neither sector would have an incentive to expand or contract. This results in an excess supply of capital that lowers the purchase price of capital relative to that of labor. The decline 6
See Chakrabarti (1998) for a generalized factor price equalization (FPE) theorem which identiBes a condition, stated in terms of the allocation of factor endowments across countries relative to the demand for and the factor intensities of goods, that is necessary and su8cient for FPE in a world with arbitrary number of countries, goods and factors. Deardor0 (1994) suggested a “lens” condition for FPE that he showed is necessary and conjectured is suCcient in a world with arbitrary number of countries, goods and factors. Chakrabarti (1994) proved that the lens condition is necessary and suCcient for FPE i7 the world has at least as many goods as factors.
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in the relative purchase price of capital induces each sector to substitute away from labor. This substitution e0ect operates against the endowment e0ect. The change in the output of each sector depends on the relative strength of the endowment e0ect and the substitution e0ect. 3. A formal analysis The analysis presented in this section builds on the structure of simple general equilibrium models developed by Jones (1965). 7 A variable with ∧ denotes the percentage change in that variable. Let aij represent the amount of factor i used to produce one unit of good j where i = K, L; LQ and KQ represent the supply of capital and labor. In addition to the two sectors producing tradable goods (X and Y ) there is a competitive sector (T ) which exclusively provides a non-tradable service that transforms capital (new or released) to Bt its use in each of the tradable goods sectors: KAj represents the amount of capital used by T exclusively for adjustment in the output of sector j; Paj is the unit cost of adjustment for sector j (i.e. the price paid by sector j to T in transforming capital to support a unit change in the output of sector j). The adjustment technology is speciBed as KAj = j dj; where
j =
(2)
+j
for dj ¿ 0;
j
for dj 6 0;
−
and j ¿ 0 are constants. All other notations used in this section are consistent with those used in Section 2. In a competitive equilibrium the zero-proBt condition requires: wLX + rKX + PaX dx = PX x;
(3a)
wLY + rKY + PaY dy = PY y;
(3b)
PaX PaY =r= : Y X
(3c)
Full employment of resources ensures: Q LX + LY = L;
(4a)
Q KX + KY + KAX + KAY = K:
(4b)
7 See also Chakrabarti (2001), Jones and Marjit (1985), Marjit (1987, 1990, 1993), Marjit and Beladi (1990), and Jones et al. (1999).
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Conditions (3a)–(4b) reduce to the following equations of change: ˆ LX wˆ + KX rˆ = Pˆ X − AX x;
(5a)
ˆ LY wˆ + KY rˆ = Pˆ Y − AY y;
(5b)
LX xˆ + LY yˆ = Lˆ + L (wˆ − r); ˆ
(5c)
ˆ ˜KX xˆ + ˜KY yˆ = Kˆ − K (wˆ − r);
(5d)
where Lj = (waLj =Pj ) and Kj = (raKj =Pj ), ij being the distributive share of factor i in industry j; Lj = Lj =L and Kj = Kj =K are fractions of the economy’s labor and K capital employed in the production of good j; Aj = ((j r)=Pj ); Aj = ((j j)=K); and K ˜ Kj = (Kj + Aj ). Let Kˆ = Lˆ = Pˆ X = Pˆ Y = 0 in the initial equilibrium. Consider the e0ect that a subsequent expansion in capital stock (i.e. Kˆ ¿ 0) has on the composition of output. Let commodity prices and labor supply remain unchanged, i.e. Lˆ = Pˆ X = Pˆ Y = 0. Under these simpliBcations, the relative change in the factor prices is given by Y yˆ − X xˆ (wˆ − r) ˆ = ; (6) || where || = [(KY − KX ) − (AX KY − AY KX )]; X = (KY + LY )AX and Y = (KX + LX )AY . It may be noted that absent any cost of adjustment (i.e. j = 0) relative factor rewards will not change in response to the rise in capital stock. Let L = (LX KX X + LY KY Y ) ¿ 0
(7a)
K = (KX LX X + KY LY Y ) ¿ 0;
(7b)
and where j is the elasticity of substitution between labor and capital in industry j. From (5c), (5d) and (6), the change in the output of each sector can be expressed as Y ˆ xˆ = K; (8a) || −X ˆ yˆ = K; (8b) || where X = (LX + (L X )=||); Y = (LY − (L Y )=||); X = (˜KX − (K X )=||); Y =(˜KY +(K Y )=||); and ||=(X Y −Y X ). (Lj =||)Kˆ captures the endowment e0ect and (L j =||)Kˆ the substitution e0ect. The endowment e0ect isolates the direct e0ect that an expansion in capital (net of the capital used in adjustment) has on the output of each sector at any given factor price ratio. The substitution e0ect arises from any change in the relative factor rewards following the excess supply of capital at the initial factor price ratio.
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Absent any cost of adjustment (i.e. j = 0) Eqs. (8a) and (8b) yield the familiar JR ˆ e0ect: (xˆ − y) ˆ = −K=|| where || = (KY − KX ) ∈ (0; 1). The capital-intensive sector expands proportionately more than the rise in the capital stock (Jones’ magniBcation e0ect) and the labor-intensive sector contracts. From (8a) and (8b), Y xˆ =− : (9) yˆ X Eq. (9) characterizes a generalized (non-linear) JR path of expansion when there are costs of adjustment. If X and Y have the same sign then one of the sectors expands at the cost of another. A condition suCcient to induce an expansion in both sectors is KY X KX ¡ : and ¡ (1 − AX ) LX L (1 − AY ) If
|X | || ¿ ; LX L
Y || ¡ LY L
and
X ¿ aKX ;
then X will contract and Y will expand. If, on the other hand, KY =(1 − AY ) ¿ KX = (1 − AX ) and Y ¿ aKY , then X will expand and Y will contract. There are interesting implications of these results for international trade. For instance, let two small economies (foreign and home) be characterized by identical endowments and preferences, share identical production technologies, and be both initially in autarky. Let the free trade price (pf ) be equal to the initial autarkic price. Let an asterisk represent the foreign country and no-asterisk the home country. Consider the possibility that adjustment technologies in the two countries exhibit the following properties: KY |∗X | | ∗ | ∗Y | ∗ | KX ¿ ; ¿ ; Y ¿ aKY ; ¿ ∗ ∗ (1 − AY ) (1 − AX ) LX ∗L LY ∗L and
X ∗ ¿ a∗KX :
Let both these economies experience an identical expansion in capital. The endowment e0ect for both these economies will be identical. The substitution e0ect for the foreign economy will reinforce the endowment e0ect and it will experience the JR e0ect: The capital-intensive sector will expand and the labor-intensive sector will contract. For the home country the substitution e0ect will operate against the endowment e0ect. If the home substitution e0ect is strong enough to more than o0set the endowment e0ect then the JR e0ect will be reversed: The labor-intensive sector will expand and the capital-intensive sector will contract. This can result in a trading equilibrium where the home country exports its labor-intensive good to the foreign country and the foreign country exports its capital-intensive good to the home country. Such a trading equilibrium between two economies that di0er in their adjustment technologies but are otherwise identical is depicted in Fig. 5, drawn under the simplifying assumption of homothetic preferences. 8 Initially production and consumption is at A for both 8 This argument provides a plausible explanation for the large volume of North–North trade relative to North–South trade.
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Fig. 5. Balanced trade between two identical countries but for an asymmetry in adjustment costs: Home country exports the labor-intensive good and foreign country exports the capital-intensive good.
countries. The production possibility frontier shifts from TT to T T (net of the capital used up in adjustment) after the expansion in capital. Production in the home country moves to P and that in the foreign country moves to P∗ . 9 At pf consumption in the home country moves to C and that in the foreign country moves to C∗ resulting in a trading equilibrium where the home country exports X (labor intensive) and the foreign country exports Y (capital intensive). 10 4. Conclusion This paper demonstrates that the introduction of asymmetric adjustment costs in a simple general equilibrium framework establishes a meaningful link between factor price determination and output determination, breaking the analytically convenient dichotomy of the H–O–S model. It is shown that in a two-sector model an expansion in the endowment of a factor (capital) can lead to an expansion, contraction or no change in the output of either sector depending on the adjustment technology. The possibility of a trading equilibrium between two economies that di0er in their adjustment technologies but are otherwise identical is demonstrated. Finally, innumerous attempts, in a 9 The slope at any point on T T reTects the ratio (MC =MC ) of marginal costs (MC) of production X Y and does not incorporate the costs of adjustment. At any given point on T T the true MC ratio (mcX =mcY ), after incorporating costs of adjustment, is higher in the foreign country than in the home country. The output mix is chosen by each country in a way that mcX =mcY is equal the free trade price ratio (pf ). 10 This argument may have implications for the Leontie0 paradox as well. For instance, in 1947, at a time when the U.S. was considered to be the most capital-rich country, the capital-labor ratio embodied in U.S. imports exceeded the ratio embodied in exports by 60% (Leontie0, 1954). This apparently “paradoxical” pattern may very well have been due to signiBcant di0erences in adjustment costs across sectors. It may also be argued that any asymmetry in costs of adjustment is a relatively short-term phenomenon and a convergence in adjustment costs over the long run can explain the fact that the Leontie0 paradox has systematically been observed to have gradually weakened, though not disappeared, in later data sets.
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vast pool of empirical literature, are continuously made to test (weak) implications of the H–O–S theory. Most studies report that the implications are spectacularly at odds with the description of international data. 11 By bringing out the implications of asymmetric adjustment costs this paper points to the importance of allowing sector-speciBc adjustment costs in tests of the implications of the H–O–S theory. References Barnett, S.A., Sakellaris, P., 1999. A new look at Brm market value, investment, and adjustment costs. The Review of Economics and Statistics 81, 250–260. Benoit, C., Serena, Ng., 1992. Adjustment costs and factor demands in Canadian manufacturing industries. Applied Economics 24, 845–857. Bernstein, J.I., 1988. Dynamic factor demands and adjustment costs: An analysis of Bell Canada’s technology. Information Economics and Policy 3, 5–24. Chakrabarti, A., 1994. Proof of suCciency of the lens condition for factor price equalization. Mimeo., University of Michigan, Ann Arbor, MI. Chakrabarti, A., 1998. A generalized factor price equalization theorem. Mimeo., University of Wisconsin, Milwaukee, WI. Chakrabarti, A., 1999. Estimation of internal adjustment costs: Capital-intensive and labor-intensive industries. Mimeo., University of Wisconsin, Milwaukee, WI. Chakrabarti, A., 2001. Rybczynski theorem and asymmetric adjustment costs. Keio Economic Studies 38, 33–41. Davis, D.R., Weinstein, D.E., Bradford, S.C., Shimpo, K., 1997. Using international and Japanese regional data to determine when the factor abundance theory of trade works. American Economic Review 87, 421– 446. Deardor0, A., 1994. The possibility of factor price equalization revisited. Journal of International Economics 47, 167–175. Helpman, E., 1999. The structure of foreign trade. Journal of Economic Perspectives 13, 121–144. Jones, R.W., 1965. The structure of simple general equilibrium models. Journal of Political Economy 73, 557–572. Jones, R.W., Marjit, S., 1985. A simple production model with Stolper–Samuelson properties. International Economic Review 26, 565–567. Jones, R.W., Marjit, S., Beladi, H., 1999. Three faces of factor intensities. Journal of International Economics 48, 413–420. Kolstad, C.D., Lee, L., 1993. The speciBcation of dynamics in cost function and factor demand estimation. The Review of Economics and Statistics 75, 721–726. Leamer, E.E., Levinsohn, J.A., 1995. International trade theory: The evidence. In: Grossman, G.M., Rogo0, K. (Eds.), Handbook of International Economics, Vol. 3. North-Holland, Amsterdam, pp. 1339 –1394. Leontie0, W.W., 1954. Domestic production and foreign trade: The American capital position re-examined. Proceedings of the American Philosophical Society 97, 332–349. Lichtenberg, F.R., 1988. Estimation of the internal adjustment costs model using longitudinal establishment data. The Review of Economics and Statistics 70, 421–430. Lucas, R., 1967. Adjustment costs and the theory of supply. Journal of Political Economy 75, 321–334. Marjit, S., 1987. Trade in intermediates and the colonial pattern of trade. Economica 54, 173–184. Marjit, S., 1990. A simple production model in trade and its applications. Economics Letters 32, 257–260. Marjit, S., 1993. Uniform tari0s in general equilibrium: A simple model. Zeitschrift f uV r Nationalokonomie 57, 189–196. 11 See TreTer (1995), Davis et al. (1997) for recent accounts of ways in which international data deviate from the theoretical predictions of the Heckscher–Ohlin model and a plausible reconciliation in terms of cross-country productivity di0erences and home bias in demand. See also Leamer and Levinsohn (1995) and Helpman (1999) for comprehensive reviews of the relevant literature.
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Marjit, S., Beladi, H., 1990. Factor underutilization and the theory of international trade. Keio Economic Studies 27, 43–51. Mohnen, P.M., Nadiri, I., Prucha, I.R., 1986. R&D, production structure, and productivity growth in the U.S., Japanese and German manufacturing sectors. European Economic Review 30, 749–771. Mulligan, C.B., Sala-i-Martin, X., 1993. Transitional dynamics in two-sector models of endogenous growth. The Quarterly Journal of Economics 108, 739–773. Mussa, M., 1978. Dynamic adjustment in the Heckscher–Ohlin–Samuelson model. Journal of Political Economy 86, 775–791. Pindyck, R., Rotemberg, J., 1983. Dynamic factor demands under rational expectations. Scandinavian Journal of Economics 85, 223–238. TreTer, D., 1995. The case of missing trade and other mysteries. American Economic Review 85, 1029–1046. Wolfson, P., 1993. Compositional change, aggregation, and dynamic factor demand: estimates on a panel of manufacturing Brms. Journal of Applied Econometrics 8, 129–148. Zanias, G.P., 1991. Adjustment costs, production functions and capital–labour substitution in Greek manufacturing: 1958–80. Applied Economics 23, 49–55.