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Privatization, Unemployment, and Welfare in the Harris-Todaro Model with a Mixed Duopoly

Jiancai Pi and Jun Yin

Abstract

This paper explores the unemployment and welfare effects of privatization through the Harris-Todaro model with a mixed duopoly. Our approach is more generalized than the existing literature. When capital is sector-specific (i. e., it is in the short run), an increase in the degree of partial privatization will raise the unemployment rate, but the change of social welfare is conditional on the market share of the public firm and the relative degree of partial privatization. When capital is sector-mobile (i. e., it is in the long run), an increase in the degree of partial privatization will reduce the unemployment rate, but the change of social welfare is also dependent on the market share of the public firm and the relative degree of partial privatization. Our results capture the fact that the public firm and the private firm usually coexist in a competitive environment in the real world.

JEL Classification: L13; L22; L33; R23

Acknowledgements:

We are very grateful to the editor Johann Brunner and two anonymous reviewers for their insightful comments on improving this paper. Pi thanks the financial support provided by the Program for New Century Excellent Talents in University, the Fundamental Research Funds for the Central Universities, and the Collaborative Innovation Center for China Economy. Any remaining errors are ours.

Appendix A: Comparative statics in the basic theoretical model

Totally differentiating eqs 1, 2 and 6 and expressing in the form of matrix, we have:

(27)s1+k1+es1s1+kesεbθKXms1+e1s1s2+e1sεbθKXmsλKXm1sλKXmsKXXˆ0Xˆ1rˆX=ks00kˆ.

Appendix B: Stability in the basic theoretical model

The dynamic adjustment of the basic theoretical model can be expressed as follows:

(28)X˙0=α1pX+kpXX0mwX,rX,
(29)X˙1=pX+pXX1mwX,rX,
(30)r˙X=α32FrwX,rX+mrwX,rXX0+X1KˉX,

where “·” indicates differentiation with respect to time, and αi (i=1,2,3) is the positive adjustment speed. Totally differentiating the above three excess demand functions, we can obtain the following Jacobian matrix:

(31)J1=s1+k1+es1s1+kesεbθKXms1+e1s1s2+e1sεbθKXmsλKXm1sλKXmsKX.

Therefore, all the principal minors of the above Jacobian matrix can be shown as:

(32)J11=s1+k1+es,
(33)J12=s1s1+k2+e,
(34)J13=J1=J12sKX+s1s1+kεbθKXmλKXm.

Following Chao, Hazari, and Yu (2006) , we can find that the sufficient condition for the stability of the system is that all the principal minors of odd order are non-positive and all the principal minors of even order are non-negative. Hence, a sufficient condition for the stability of the system is given, namely e>1. It is easy to find that Δ1=J1, thus we have Δ1<0.

Appendix C: Proof of the sign of eq. [12]

Substituting eqs 8, 9, and 10] into 12, we have:

(35)wˆYkˆ=1+μΦkss1sλLXmsKX+λKXmsLXΔ1<0.

Appendix D: Proof of Proposition 2

We set k=μwYLXμˆ/kˆpX0XXˆ/kˆ, and then 15 can be reduced to:

(36)Vˆkˆ=kkμwYLXVkμˆkˆ.

When s<s, from Proposition 1 we know that Xˆkˆ<0. Combining with the results that p<0 and μˆkˆ>0, we have k<0. Recall that k0,1, and we can obtain k>k and Vˆkˆ<0.

When s>s, from Proposition 1 we know that Xˆkˆ>0. Combining with the results that p<0 and μˆkˆ>0, we have k>0. In the case where k1, recall that k0,1, and we can get k<k and Vˆkˆ<0. In the case where k<1, if k<k, then Vˆkˆ<0; and if k>k, then Vˆkˆ>0.

Appendix E: Comparative statics in the extended model

Totally differentiating eqs 16, 17, 18, 19, and 20 with the help of differentiated 4, and expressing in the form of matrix, we have:

(37)s1+k1+es1s1+kes00εbθKXms1+e1s1s2+e1s00εbθKXm000θLYθKY1+μsλLXm1+μ1sλLXmλLY1+μλLX+sLY1+μsLX+sLYsλKXm1sλKXmλKYsKYsKX+sKYXˆ0Xˆ1YˆwˆYrˆ=ks0000kˆ.

Appendix F: Stability in the extended model

The dynamic adjustment of the extended model can be expressed as follows:

(38)X˙0=β1pX+kpXX0mwX,r,
(39)X˙1=β2pX+pXX1mwX,r,
(40)Y˙=β31gwY,r,
(41)w˙Y=β41+μ2FwwX,r+mwwX,rX0+X1+gwwY,rYLˉ,
(42)r˙=β52FrwX,r+mrwX,rX0+X1+grwY,rYKˉ,

where βii=1,2,,5 is the positive adjustment speed. Totally differentiating the above five excess demand functions, we can get the following Jacobian matrix:

(43)J2=s1+k1+es1s1+kes00εbθKXms1+e1s1s2+e1s00εbθKXm000θLYθKY1+μsλLXm1+μ1sλLXmλLY1+μλLX+sLY1+μsLX+sLYsλKXm1sλKXmλKYsKYsKX+sKY.

Therefore, all the principal minors of the above Jacobian matrix can be shown as:

(44)J21=s1+k1+es,
(45)J22=s1s1+k2+e,
(46)J23=0,
(47)J24=θLYλLYJ22,
(48)J25=J2=J22Ψ+s1s1+kεbθLYθKXmλ.

According to Chao, Hazari, and Yu (2006) , we also introduce a new sufficient condition for the stability of the two-sector mobile-capital system, namely e>1 and λ<0. It is easy to find that Δ2=J2, thus we have Δ2<0.

Appendix G: Proof of Proposition 4

We set k=μwYLXμˆ/kˆpX0XXˆ/kˆ, and then 15 can be reduced to:

(49)Vˆkˆ=kkμwYLXVkμˆkˆ.

When s>s, from Proposition 3 we know that Xˆkˆ>0. Combining with the results that p<0 and μˆkˆ<0, we have k<0. Recall that k0,1, and we can obtain k>k and Vˆkˆ>0.

When s<s, from Proposition 3 we know that Xˆkˆ<0. Combining with the results that p<0 and μˆkˆ<0, we have k>0. In the case where k1, recall that k0,1, we can get k<k and Vˆkˆ>0. In the case where k<1, if k<k, then Vˆkˆ>0; and if k>k, then Vˆkˆ<0.

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Published Online: 2016-11-18
Published in Print: 2016-10-1

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