Journal of International Trade & Economic Development ISSN: 09638199 (Print) 14699559 (Online) Journal homepage: https://www.tandfonline.com/loi/rjte20 Rising wage inequality in developing economies: Privatization and competition ChiChur Chao , Bharat R. Hazari & Eden S. H. Yu To cite this article: ChiChur Chao , Bharat R. Hazari & Eden S. H. Yu (2006) Rising wage inequality in developing economies: Privatization and competition, Journal of International Trade & Economic Development, 15:3, 375385, DOI: 10.1080/09638190600871719 To link to this article: https://doi.org/10.1080/09638190600871719 Published online: 18 Feb 2007. Submit your article to this journal Article views: 128 View related articles Citing articles: 4 View citing articles Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=rjte20 J. Int. Trade & Economic Development Vol. 15, No. 3, 375 – 385, September 2006 Rising Wage Inequality in Developing Economies: Privatization and Competition CHICHUR CHAO*, BHARAT R. HAZARI** & EDEN S. H. YU** *Department of Economics, Chinese University of Hong Kong, Hong Kong and Department of Economics, Oregon State University, USA, **Department of Economics and Finance, City University of Hong Kong, Hong Kong ABSTRACT Using a dual structure depicting a developing economy, this paper shows that increased partial privatization or foreign competition can lead to wage inequality between skilled and unskilled labor. In addition, rising wage inequality can be triggered by inﬂows of unskilled labor or outﬂows of skilled labor and/or capital. Further, partial privatization or foreign competition reduces the urban output, thereby raising the goods price and unemployment ratio. These eﬀects lower social welfare of the economy. KEY WORDS: Wage inequality, privatization, competition, developing economies Introduction The ‘Iron Lady’ Baroness M. Thatcher is the mother of moves towards privatization throughout the world. In many countries, even supposedly left wing parties sold public assets to the private sector. The labor government in Australia privatized banks and airlines. The Marxist government in Bengal India also privatized many public assets. In addition to privatization, the world has also witnessed the delivery of a more liberal market oriented agenda in many countries after the collapse of the Communist regimes. The rise of globalization and liberal reforms has been closely connected with trade reforms in many developed and developing countries. In spite of such movements towards privatization and trade reforms, very few studies have been undertaken to assess the impact of such movements on wage inequalities. It should be noted that both privatization and trade reform alter the rules by which ﬁrms are managed and governed and also have an impact on the nature of markets in which ﬁrms operate. Correspondence Address: Bharat R. Hazari, Department of Economics and Finance, City University of Hong Kong, Kowloon, Hong Kong. Email: [email protected] ISSN 09638199 Print/14699559 Online Ó 2006 Taylor & Francis DOI: 10.1080/09638190600871719 376 C.C. Chao et al. Wage inequality between skilled and unskilled labor has been the focus of attention of many studies. In the literature, the skill premium can be primarily attributed to trade and technology:1 the increase in the wages of the skilled workers being associated with skilledbiased technological progress, while the falling wage of the unskilled workers is linked to cheaper imports of unskilledlaborintensive goods or outsourcing of unskilledlaborintensive stages abroad.2 The analyses of skill premium mainly focus on product restructuring towards skillbiased technologies, often noticed in advanced economies. Nonetheless, for developing economies, rising wage inequality may also be caused by a diﬀerent set of factors, such as distortions, both autonomous and/or policy induced. Recently, some studies have emerged on skill formation and wage inequality for developing economies; for example, Davis (1998) attributes wage inequality to unemployment, Kar and Beladi (2004) to migration, and Feenstra and Hanson (1997) and Marjit et al. (2003) to foreign direct investment. These studies mainly concentrate on market distortions and the associated factor mobility. However, the impacts of economic policies and reforms on relative wages of the skilled and unskilled workers in developing countries have by and large not been addressed. In this paper we address these issues by focusing on the following two questions: does privatization cause rising wage inequality in developing countries? And does increased foreign competition increase relative wage dispersion?3 This paper is organized as follows. The next section provides a model of a dual economy relevant to many developing economies, in which a state owned monopolistic ﬁrm operates in the urban sector, while the ﬁrms in the rural sector are competitive and privately owned. Using this model, the eﬀects of privatization and competition on the wages of the skilled and unskilled are examined in the third section. This is followed by an analysis of welfare implications of such reforms in the fourth section. The ﬁnal section contains concluding remarks. The Model We consider a developing economy that consists of two regions: urban and rural. A partially stateowned ﬁrm in the urban area produces manufac turing good X, while competitive privatelyowned ﬁrms in the rural area produce the agricultural good Y. The country exports good Y and imports good X. There are no impediments for the exports, but to protect the state owned ﬁrm the government imposes a quota Q of good X on foreign exporters. Choosing good Y as the numeraire, the domestic relative price of good X is denoted by p. Denoting domestic consumers’ demand for the two goods by DX and DY, the utility function is represented by the following quasilinear utility func tion: U(DX, DY) ¼ u(DX) þ DY, where u0 (DX) 4 0. Letting I denote income, utility maximization subject to the budget constraint, I ¼ pDX þ DY, yields Rising Wage Inequality in Developing Economies 377 the inverse demand function for good X: p ¼ p(DX), with p0 5 0, and the indirect utility function is: V ¼ V(p, I), with Vp ¼ 7DX and VI ¼ 1. In equilibrium, demand for good X meets its supply from domestic production and imports, i.e. DX ¼ X þ Q. It is appropriate now to examine the production side of the economy. In the urban sector, the stateowned monopolistic ﬁrm produces good X under increasing returns to scale: skilled labor along with capital is required for the needed ﬁxed inputs, while unskilled labor and capital are used as variable inputs in production. Letting F( ) and m( ) be the ﬁxed and marginal costs, total cost for producing good X is: C(wU, wS, r, X) ¼ F(wS, r) þ m(wU, r)X, where wU and wS are respectively the urban unskilled and skilled wages, and r is the capital rental rate. By the envelope property, the employment of skilled labor, unskilled labor and capital in sector X are: S ¼ Fw(wS, r), LX ¼ mw(wU, r)X and KX ¼ Fr(wS, r) þ mr(wU, r)X, where the subscript represents the partial derivative. Since the ﬁrm in sector X is stateowned, it cares not only for proﬁts but also for the welfare generated. The proﬁt of ﬁrm X is: p ¼ p(X þ Q)X – C(wU, wS, r, X), and welfare is the sum of the proﬁt and consumer surplus: W ¼ p þ CS, where CS ¼ u(X þ Q) – p(X þ Q)X. Therefore, the objective of the ﬁrm is to maximize a weighted average of proﬁts and welfare, kp þ (1 – k)W, where k 2 [0, 1].4 Here, the weight k represents the degree of private ownership: the ﬁrm is fully stateowned when k ¼ 0, whereas it is completely privateowned when k ¼ 1. Hence, a larger value of k denotes a higher level of private ownership.5 The ﬁrm chooses the level of output X to maximize the objective function already postulated, and the ﬁrstorder optimality condition is: pðX þ QÞ þ kp0 ðX þ QÞX ¼ mðwU ; rÞ ð1Þ where the two terms on the lefthand side show the marginal revenue (MR) of producing good X. Equation (1) is interpreted as follows. For the fully stateowned ﬁrm (k ¼ 0), marginal cost pricing follows from equation (1) and the ﬁrm behaves competitively (i.e. p ¼ m). When partial privatization takes place (k 4 0), we have p 4 m because p0 () 5 0. This causes the MR curve to shift downward for a larger k. In addition, the direct eﬀect of a rise in Q on MR is: @MR/@Q ¼ (1 þ kes)p0 , where s ¼ X/(X þ Q) 5 1 and e ¼ (X þ Q)p00 /p0 > < 0. By the stability condition, 1 þ es 4 0 as shown in the appendix, we have @MR/@Q 5 0. This implies that increased foreign competition due to a relaxation of quotas also causes the MR curve of the domestic ﬁrm to shift downward.6 Besides the welfare consideration, the stateowned urban ﬁrm also sets, for unskilled labor only, a minimum wage, which exceeds the unskilled labor wage in the rural sector. The higher urban wage leads to rural urban migration of unskilled workers, thereby resulting in urban unemploy ment, Lu. Deﬁning the unemployment ratio of unskilled workers in 378 C.C. Chao et al. the urban sector by m ¼ Lu/LX, the Harris – Todaro (1970) migration equili brium is: wU =ð1 þ mÞ ¼ wR ; ð2Þ where wR is the rural unskilled wage and 1/(1 þ m) expresses the probability to ﬁnd a job by unskilled workers in the urban area. Equation (2) simply states that in equilibrium, the expected urban unskilled wage equals the rural unskilled wage. As far as the production of good Y is concerned, it requires unskilled labor and capital under constant returns to scale. The corresponding unit cost function is denoted by g(wR, r), and the demands for unskilled labor and capital are: LY ¼ gw(wR, r)Y and KY ¼ gr(wR, r)Y. Under perfect com petition, unit cost equals the price of good Y: gðwR ; rÞ ¼ 1 ð3Þ where the price of good Y is normalized to unity. Consider next the factor markets. The equilibrium conditions for skilled labor, unskilled labor and capital are: Fw ðwS ; rÞ ¼ S ð4Þ ð1 þ mÞmw ðwU ; rÞX þ gw ðwR ; rÞY ¼ L ð5Þ Fr ðwS ; rÞ þ mr ðwU ; rÞX þ gr ðwR ; rÞY ¼ K ð6Þ where S, L and K denote the amounts of skilled labor, unskilled labor and capital available in the economy. Equations (1) – (6) describe the dual developing economy, which consists of six unknowns, wR, wS, r, m, X and Y. We will use this model to study the eﬀects of partial privatization and foreign competition on the wages of skilled and unskilled labor. In addition, the implications on social welfare will be examined. Wage Inequality In this paper, partial privatization is represented by a rise in the weight k towards proﬁt maximization, while foreign competition is reﬂected by an increase in the quota on foreign exports Q. Due to monopoly in sector X, changes in k or Q aﬀect the marginal revenue of producing good X and hence the return to capital as shown by equation (1). Letting a circumﬂex represent a percentage change, these eﬀects can be obtained by totally diﬀerentiating equation (1): ebym r ¼ ks k^ ð1 sÞð1 þ kesÞQ KX ^ ^ s½1 þ kð1 þ esÞX^ ð7Þ Rising Wage Inequality in Developing Economies 379 where e ¼ 7p/(X þ Q)p0 4 0, being the price elasticity of demand for good X. Note that b ¼ m/p 5 1 and ym jX represents the distributive share of the jth factor in the X’s variable cost. As indicated by changes in k or Q and X on the righthand side of equation (7), partial privatization or foreign competition not only directly but also indirectly inﬂuences the marginal revenue of producing good X. The former causes the MR curve to shift downward, while the latter leads to a movement along the curve via the change in X. Consequently, given the urban unskilled wage, the change in the capital return in equation (7) reﬂects the change in MR. This then aﬀects the wages of unskilled and skilled labor according to equations (3) and (4), as follows: ^R ¼ ðyKY =yLY Þ^r w ð8Þ w ^ F ^S ¼ r^ S=s ð9Þ SX where yjY represents the distributive share of the jth factor in industry Y and sFSX expresses the substitution eﬀect in the demand for skilled labor in sector X.7 In addition, the change in the rural unskilled wage causes rural – urban migration, thereby aﬀecting the urban unemployment ratio by equation (2): ^ ¼ ½ð1 þ mÞ=mw m ^R ð10Þ To get the overall changes in the skilled and unskilled wages, we need to obtain the eﬀects of partial privatization or foreign competition on the output of good X. Totally diﬀerentiating the resource constraints in equations (4) and (5), we have: ð1 þ mÞlm ^ ^ ^ m LX X þ lLY Y ¼ L ½ð1 þ mÞsLX þ sLY ^ ^R mlm r þ sLY w LX m ^ ð11Þ lm ^ ^ ^ m KX X þ lKY Y ¼ K þ ðsKX þ sKY Þ^ ^R sFKX ðw r sKY w ^S r^Þ ð12Þ where lmjX and ljY are respectively the employment shares of factor j in industry X’s and industry Y’s variable inputs. In addition, sji captures the eﬀect of a change in the factor prices on the demand for factor j in sector i.8 Substituting expressions from equations (7) – (10) into (11) and (12), we can solve for the eﬀects of partial privatization or foreign competition on the output of good X: ^ k^ ¼ ksA=D < 0 X= ð13Þ ^ Q X= ^ ¼ ð1 sÞð1 þ kesÞA=D < 0 ð14Þ where D ¼ s½1 þ kð1 þ esÞA ebyLY ym 9 KX l > 0 by stability and A 4 0. Note m m that l ½¼ ð1 þ mÞlLX lKY lKX lLY is positive (negative) if industry X is 380 C.C. Chao et al. labor intensive relative to industry Y in variable inputs. Hence, regardless of the factor intensity, partial privatization or foreign competition exerts a pressure on the output of good X and the resulting impact on the its price can be obtained by diﬀerentiating the inverse demand function of good X, i.e. p ¼ p(X þ Q): p^=k^ ¼ ðs=eÞðX= ^ >0 ^ kÞ ð15Þ ^ ¼ ð1=eÞ½1 s þ sðX= p^=Q ^ > ^ QÞ <0 ð16Þ Equation (15) states that partial privatization reduces domestic production of good X, thereby raising its price. However, in equation (16), depending on the increase in foreign exports relative to the fall in domestic production, the price eﬀect of foreign competition is ambiguous. Nonetheless, increased foreign competition can raise the domestic price of good X if its domestic supply falls (i.e. 7dX/dQ 4 1).10 We are now ready to examine the eﬀect of increased partial privatization or foreign competition on the wages of skilled and unskilled labor. From equation (7), partial privatization or foreign competition in sector X immediately causes a downward shift of its MR curve, and then the induced fall in the production of good X raises the marginal revenue along the MR curve. The total change in MR can be traced in Figure 1 from point a to b and then to point c. The higher MR at point c gives a higher return to capital, which leads to a higher wage for skilled labor but a lower wage for rural unskilled labor. These results on the changes in wages can be obtained by substituting expressions from equations (13) and (14) into equations (7) – (9): ^R =k^ ¼ ks yKY l=D w ð17Þ ^S =k^ ¼ ðyLY =yKY Þðw w ^ ^R =kÞ ð18Þ ^ ¼ ð1 sÞð1 þ kesÞyKY l=D ^ R =Q w ð19Þ ^ ¼ ðyLY =yKY Þðw ^ S =Q w ^ ^R =QÞ ð20Þ Increased partial privatization or foreign competition can lead to a rise in the skilled wage but a fall in the rural unskilled wage, if the urban sector is unskilled intensive relative to the rural sector in variable inputs. In addition, by equation (10), the fall in the rural unskilled wage results in rural to urban migration, thereby giving rise to a higher urban unemployment ratio: ^=k^ ¼ ½ð1 þ mÞ=mðw m ^ ^R =kÞ ð21Þ m ^ ¼ ½ð1 þ mÞ=mðw ^ =Q ^ ^R =QÞ ð22Þ Rising Wage Inequality in Developing Economies 381 Figure 1. Privatization or competition on factor returns Note that rising wage inequality for the developing economy can be also triggered by the changes in factor endowments. From equations (7) – (12), we have: ^R =L^ ¼ s½1 þ kð1 þ esÞlKY yKY =D < 0 w ð23Þ ^S =L^ ¼ ðyLY =yKY Þðw w ^ >0 ^R =LÞ ð24Þ ^R =S^ ¼ s½1 þ kð1 þ esÞlLY yKY ðsFKX =sFSX Þ=D > 0 w ð25Þ ^S =S^ ¼ ðyLY =yKY Þðw w ^ <0 ^R =SÞ ð26Þ ^R =K^ ¼ s½1 þ kð1 þ esÞlLY yKY =D > 0 w ð27Þ ^S =K^ ¼ ðyLY =yKY Þðw w ^ <0 ^R =KÞ ð28Þ Therefore, from these generalized expressions, which take account of state ownership via the presence of k, either inﬂows of unskilled labor or 382 C.C. Chao et al. outﬂows of skilled labor and/or capital aggravate wage inequality in the economy. In summary, we have the following results. Proposition 1 For a developing economy with a stateowned ﬁrm in the urban sector, increased partial privatization or foreign competition can result in rising wage inequality between the skilled and unskilled labor. In addition, changes in the endowments, such as inﬂows of unskilled labor or outﬂows of skilled labor and/or capital, can also give rise to the wage inequality in the economy. Social Welfare We consider next the welfare implications of partial privatization and foreign competition for the developing economy, in which social welfare can be represented by the indirect utility function, V ¼ V(p, I), where I is national income. Note that national income comes from factor income plus proﬁts of the urban stateowned ﬁrm: I ¼ wULX þ wRLY þ wSS þ rK þ p. Totally diﬀerentiating the indirect utility function and then using equations (1) – (6), we obtain the change in social welfare: dV ¼ ðp mÞdX Qdp wR LX dm ð29Þ Equation (29) captures the existence of three distortions in the economy: monopoly power of the stateowned ﬁrm, quota restriction on foreign ﬁrms, and unemployment in the urban sector. Lowering the distortions can improve social welfare of the economy. Moreover, trade oﬀs can also be worked out. First, the eﬀect of partial privatization on welfare can be obtained from equation (29) as follows: dV=dk ¼ ðp mÞðdX=dkÞ Qðdp=dkÞ wR LX ðdm=dkÞ ð30Þ Since dX/dk 5 0, dp/dk 4 0 and dm/dk 4 0 by equations (13), (15) and (21), we have dV/dk 5 0 in equation (30). Partial privatization unambiguously reduces social welfare because it worsens the monopoly power of the state owned ﬁrm, increases the price of good X and raises the unemployment ratio in the urban sector. Secondly, from equation (29), the welfare eﬀect of increased foreign competition is: dV=dQ ¼ ðp mÞðdX=dQÞ Qðdp=dQÞ wR LX ðdm=dQÞ ð31Þ Although foreign competition reduces domestic output [dX/dQ 5 0 in equation (14)] and raises urban unemployment ratio [dm/dQ 4 0 in Rising Wage Inequality in Developing Economies 383 equation (22)], the welfare eﬀect of foreign competition is, in general, ambiguous because dp/dQ > < 0 in equation (16). However, if the increased imports crowd out more domestic production of good X (i.e. 7dX/dQ 4 1), we have dp/dQ 4 0 in equation (16) and hence dV/dQ 5 0 in equation (31). The following proposition is immediate. Proposition 2 For a developing economy with a stateowned ﬁrm in the urban sector, increased partial privatization or foreign competition can lead to a decrease in the urban output, thereby resulting in the higher goods price and unemploy ment ratio in the urban area. These eﬀects adversely impact on social welfare in the economy. Conclusions Using a dual structure for depicting a developing economy, this paper has examined the impacts of partial privatization and foreign competition on wage inequality and social welfare. Increased partial privatization or foreign competition can lead to wage inequality between the skilled and unskilled labor. In addition, rising wage inequality can be caused by changes in factor endowments such as inﬂows of unskilled labor or outﬂows of skilled labor and capital. Further, partial privatization or foreign competition can reduce the urban output, thereby raising the goods price and unemployment ratio in the urban area. These eﬀects lower social welfare of the economy. The employment and eﬃciency issues of privatization are not addressed in this paper. Although empirical studies ﬁnd mixed results on the eﬀects of privatization on employment, production eﬃciency and proﬁts,11 it may be worthwhile to consider for future research these issues in the general equilibrium analysis of privatization.12 Acknowledgement We are indebted to an anonymous referee for useful comments. The usual disclaimer applies. The work described in this paper was supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. CUHK4110/04H). Notes 1 The trade argument for rising wage inequality can be found in Wood (1995) and Leamer (1998), while the technology factor is provided in Francois and Nelson (1998). 2 See Feenstra and Hanson (1996). 3 Neary (2002) examines the eﬀect of increased foreign competition on wage inequality for advanced economies, in which ﬁrms are privately owned. 384 C.C. Chao et al. 4 This objective function of proﬁts and welfare for the partially privatized ﬁrm is adapted from Matsumura (1998), and is utilized by Chao and Yu (2006) to examine the eﬀect of privati zation on optimum tariﬀ. 5 For generating some sort of consumer surplus, proﬁts of the public enterprise may be negative. However, public ﬁrms can be proﬁt making. Telstra in Australia is a good example of partial privatization: it does not charge monopoly prices and follows the strategy of privatization. This is because the Australian government is very concerned with ensuring services to the rural region and consumers do not pay for telephone services at the true cost of providing them. 6 See Chao and Yu (1997) for a related study. 7 Since the ﬁxed cost of sector X is F(wS, r), the elasticity of factor substitution between skilled labor and capital is sFX ¼ FFwr =Fw Fr . Following Jones (1965), the substitution eﬀect in skilled labor demand is: sFSX ¼ sFX yFKX , where yFKX ð¼ rFr =FÞ is the distributive share of capital in the ﬁxed cost of sector X. 8 For instance, sLY ¼ sYyKYlLY, where sY ¼ ggwr/gwgr. 9 m We deﬁne A as: A ¼ lLY sKY þ lKY sLY þ yLY lLY sm KX þ ð1 þ mÞlKY lLX . 10 From equation (14), we can write: dX=dQ ¼ ð1 þ kesÞ=½1 þ kes þ ðk eb yLY ym > KX l=sAÞ < 1 if ks > < eb yLY ym KX l=A. Here, the positive l (i.e. the urban sector is relatively unskilled intensive in variable inputs) is a necessary condition for 7dX/dQ 4 1. 11 For example, Oum et al. (2000) show that the eﬀect of privatization on production eﬃciency is weak for the airline industry. Using a large crosssectional sample, Dewenter and Malatesta (2001) also report poor longrun performance of corporations after privatization. 12 We thank for the referee for pointing out that employment, rather than proﬁts, may be another consideration for the public ﬁrm. References Chao, C. C. and Yu, E. S. H. (1997) Trade liberalization in oligopolistic competition with unemployment: a general equilibrium analysis, Canadian Journal of Economics, 30, pp. 479 – 496. Chao, C. C. and Yu, E. S. H. (2006) Partial privatization, foreign competition, and optimum tariﬀ, Review of International Economics, 14, pp. 97 – 102. Davis, D. R. (1998) Technology, unemployment and relative wages in a global economy, European Economic Review, 42, pp. 1613 – 1633. Dewenter, K. L. and Malatesta, P. H. (2001) Stateowned and privately owned ﬁrms: an empirical analysis of proﬁtability, leverage, and labor intensity, American Economic Review, 91, pp. 320 – 334. Feenstra, R. C. and Hanson, G. H. (1996) Globalization, outsourcing, and wage inequality, American Economic Review, 86, pp. 240 – 245. Feenstra, R. C. and Hanson, G. H. (1997) Foreign direct investment and relative wages: evidence from Mexico’s maquiladoras, Journal of International Economics, 42, pp. 371 – 393. Francois, J. F. and Nelson, D. (1998) Trade, technology, and wages: general equilibrium mechanics, Economic Journal, 108, pp. 1483 – 1499. Kar, S. and Beladi, H. (2004) Skill formation and international migration: welfare perspective of developing countries, Japan and the World Economy, 16, pp. 35 – 54. Leamer, E. E. (1998) In search of Stolper – Samuelson linkages between international trade and lower wages, in: S. Collins (Ed.) Imports, Exports and the American Worker (Washington, DC: Brookings Institution). Harris, J. R. and Todaro, M. (1970) Migration, unemployment and development: a twosector analysis, American Economic Review, 60, pp. 126 – 142. Jones, R. W. (1965) The structure of simple general equilibrium models, Journal of Political Economy, 73, pp. 557 – 572. Rising Wage Inequality in Developing Economies 385 Marjit, S., et al. (2003) Trade and wage inequality in developing countries, Economic Inquiry, 42, pp. 295 – 303. Matsumura, T. (1998) Partial privatization in mixed duopoly, Journal of Public Economics, 70, pp. 473 – 483. Neary, J. P. (2002) Foreign competition and wage inequality, Review of International Economics, 10, pp. 680 – 693. Oum, T. H. et al. (2000) Socially optimal capacity and capital structure in oligopoly: the case of the airline industry, Journal of Transport Economics and Policy, 34, pp. 55 – 68. Wood, A. (1995) How trade hurt unskilled workers, Journal of Economic Perspectives, 9, pp. 57 – 80. Appendix Letting a dot over a variable denote the time derivative, the adjustments of the system in equations (1) – (6) can be expressed as: 0 1 0 10 1 X_ s½1 þ kð1 þ esÞ 0 0 ebym KX X^ B Y_ C B 0 0 yLY yKY CB ^C B C¼B CB Y C @w_ RA @ ð1 þ mÞlm m m A@ lLY ½ð1 þ mÞlLX þ sLY ð1 þ mÞsLX þ sLY w ^RA LX m m r_ lKX lKY sKY ðsKX þ sKY Þ ^r The principal minors of the above coeﬃcient matrices are, as follows: D1 ¼ s½1 þ kð1 þ esÞ D2 ¼ 0 D3 ¼ s½1 þ kð1 þ esÞlLY yLY D4 ¼ D The stability condition requires that the odd principal minors are nonpositive and the even principal minors are nonnegative. Hence, we need: (i) 1 þ es 4 0; and (ii) D 4 0 for stability. Note that 1 þ kes 4 0 since 1 þ es 4 0 and 0 k 1.
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