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

  • Jiancai Pi EMAIL logo and Jun Yin
Published/Copyright: November 18, 2016
The B.E. Journal of Economic Analysis & Policy
From the journal The B.E. Journal of Economic Analysis & Policy Volume 16 Issue 4

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.

Keywords: Harris-Todaro model; mixed duopoly; privatization; unemployment; welfare
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+es−1−s1+kes−εbθKXm−s1+e1−s−1−s2+e1−s−εbθKXmsλKXm1−sλKXm−sKXXˆ0Xˆ1rˆX=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+kp′X⋅X0−mwX,rX,
(29)X˙1=pX+p′X⋅X1−mwX,rX,
(30)r˙X=α32FrwX,rX+mrwX,rX⋅X0+X1−KˉX,

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+es−1−s1+kes−εbθKXm−s1+e1−s−1−s2+e1−s−εbθKXmsλKXm1−sλKXm−sKX.

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

(32)J11=−s1+k1+es,
(33)J12=s1−s1+k2+e,
(34)J13=J1=−J12sKX+s1−s1+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+μΦ⋅kss1−sλLXmsKX+λKXmsLXΔ1<0.

Appendix D: Proof of Proposition 2

We set k∗=−μwYLXμˆ/kˆp′X0XXˆ/kˆ, and then 15 can be reduced to:

(36)Vˆkˆ=k−k∗μwYLXVk∗μˆkˆ.

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 k∈0,1, 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 k∗≥1, recall that k∈0,1, 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+es−1−s1+kes00−εbθKXm−s1+e1−s−1−s2+e1−s00−εbθKXm000−θLY−θKY1+μsλLXm1+μ1−sλLXmλLY−1+μλLX+sLY1+μsLX+sLYsλKXm1−sλKXmλKYsKY−sKX+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+kp′X⋅X0−mwX,r,
(39)X˙1=β2pX+p′X⋅X1−mwX,r,
(40)Y˙=β31−gwY,r,
(41)w˙Y=β41+μ2FwwX,r+mwwX,r⋅X0+X1+gwwY,r⋅Y−Lˉ,
(42)r˙=β52FrwX,r+mrwX,r⋅X0+X1+grwY,r⋅Y−Kˉ,

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+es−1−s1+kes00−εbθKXm−s1+e1−s−1−s2+e1−s00−εbθKXm000−θLY−θKY1+μsλLXm1+μ1−sλLXmλLY−1+μλLX+sLY1+μsLX+sLYsλKXm1−sλKXmλKYsKY−sKX+sKY.

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

(44)J21=−s1+k1+es,
(45)J22=s1−s1+k2+e,
(46)J23=0,
(47)J24=θLYλLYJ22,
(48)J25=J2=J22Ψ+s1−s1+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/kˆp′X0XXˆ/kˆ, and then 15 can be reduced to:

(49)Vˆkˆ=k−k∗∗μwYLXVk∗∗μˆkˆ.

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 k∈0,1, 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 k∗∗≥1, recall that k∈0,1, 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

©2016 by De Gruyter

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https://doi.org/10.1515/bejeap-2016-0016
Keywords for this article
Harris-Todaro model; mixed duopoly; privatization; unemployment; welfare

Articles in the same Issue

  1. Research Article
  2. Voting in Central Banks: Theory versus Stylized Facts
  3. Delinquency Reinforcement and Balance: Is Exposure to Delinquent Peers Always Risky?
  4. Can Polluting Firms Favor Regulation?
  5. Cannabis Control and Crime: Medicinal Use, Depenalization and the War on Drugs
  6. Coasean Quality of Regulated Goods
  7. Can Low-Wage Employment Help People Escape from the No-Pay – Low-Income Trap?
  8. Auctioning Emission Permits with Market Power
  9. Privatization, Unemployment, and Welfare in the Harris-Todaro Model with a Mixed Duopoly
  10. Does Eco-labeling of Services Matter? Evidence from Higher Education
  11. What Do Regulators Value?
  12. Strategic CSR, Heterogeneous Firms and Credit Constraints
  13. Regulations to Supplement Weak Environmental Liability
  14. Intergenerational Educational Persistence among Daughters: Evidence from India
  15. A Signal of Altruistic Motivation for Foreign Aid
  16. Has Creative Destruction become more Destructive?
  17. Is There a Role for Higher Education Institutions in Improving the Quality of First Employment?
  18. Contribution
  19. The Effects of School Closure Threats on Student Performance: Evidence from a Natural Experiment
  20. School Entry, Compulsory Schooling, and Human Capital Accumulation: Evidence from Michigan
  21. Topics
  22. Overeducation, Overskilling and Mental Well-being
  23. Process and Product Innovation and the Role of the Preference Function
  24. Letter
  25. Does Evasion Invalidate the Welfare Sufficiency of the ETI?
  26. How Lobbying Affects Representation: Results for Majority-Elected Politicians
  27. Consumers’ Misevaluation and Public Promotion
  28. Meet-the-competition clauses and the strategic disclosure of product quality
  29. Import Competition and Post-displacement Wages in Korea: Whom You Trade with Matters
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