IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v7y2019i2p49-d227500.html
   My bibliography  Save this article

Stackelberg Equilibrium Premium Strategies for Push-Pull Competition in a Non-Life Insurance Market with Product Differentiation

Author

Listed:
  • Søren Asmussen

    (Department of Mathematics, Aarhus University, 8000 Aarhus, Denmark)

  • Bent Jesper Christensen

    (Department of Economics and Business Economics and the Dale T. Mortensen Centre, Aarhus University, 8210 Aarhus, Denmark)

  • Julie Thøgersen

    (Department of Mathematics, Aarhus University, 8000 Aarhus, Denmark)

Abstract

Two insurance companies I 1 , I 2 with reserves R 1 ( t ) , R 2 ( t ) compete for customers, such that in a suitable differential game the smaller company I 2 with R 2 ( 0 ) < R 1 ( 0 ) aims at minimizing R 1 ( t ) − R 2 ( t ) by using the premium p 2 as control and the larger I 1 at maximizing by using p 1 . Deductibles K 1 , K 2 are fixed but may be different. If K 1 > K 2 and I 2 is the leader choosing its premium first, conditions for Stackelberg equilibrium are established. For gamma-distributed rates of claim arrivals, explicit equilibrium premiums are obtained, and shown to depend on the running reserve difference. The analysis is based on the diffusion approximation to a standard Cramér-Lundberg risk process extended to allow investment in a risk-free asset.

Suggested Citation

  • Søren Asmussen & Bent Jesper Christensen & Julie Thøgersen, 2019. "Stackelberg Equilibrium Premium Strategies for Push-Pull Competition in a Non-Life Insurance Market with Product Differentiation," Risks, MDPI, vol. 7(2), pages 1-23, May.
    • Handle: RePEc:gam:jrisks:v:7:y:2019:i:2:p:49-:d:227500
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/7/2/49/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-9091/7/2/49/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Emms, Paul & Haberman, Steven, 2005. "Pricing General Insurance Using Optimal Control Theory," ASTIN Bulletin, Cambridge University Press, vol. 35(2), pages 427-453, November.
    2. Steven Salop & Joseph Stiglitz, 1977. "Bargains and Ripoffs: A Model of Monopolistically Competitive Price Dispersion," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 44(3), pages 493-510.
    3. Mortensen, Dale & Pissarides, Christopher, 2011. "Job Creation and Job Destruction in the Theory of Unemployment," Ekonomicheskaya Politika / Economic Policy, Russian Presidential Academy of National Economy and Public Administration, vol. 1, pages 1-19.
    4. Mattias K. Polborn, 1998. "A Model of an Oligopoly in an Insurance Market," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 23(1), pages 41-48, June.
    5. Michael Rothschild & Joseph Stiglitz, 1976. "Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 90(4), pages 629-649.
    6. Jin, Zhuo & Yin, G. & Wu, Fuke, 2013. "Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 733-746.
    7. Pantelous, Athanasios A. & Passalidou, Eudokia, 2015. "Optimal premium pricing strategies for competitive general insurance markets," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 858-874.
    8. Meng, Hui & Li, Shuanming & Jin, Zhuo, 2015. "A reinsurance game between two insurance companies with nonlinear risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 91-97.
    9. Pantelous, Athanasios A. & Passalidou, Eudokia, 2017. "Optimal strategies for a non-linear premium-reserve model in a competitive insurance market," Annals of Actuarial Science, Cambridge University Press, vol. 11(1), pages 1-19, March.
    10. Emms, P. & Haberman, S. & Savoulli, I., 2007. "Optimal strategies for pricing general insurance," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 15-34, January.
    11. Boonen, Tim J. & Pantelous, Athanasios A. & Wu, Renchao, 2018. "Non-cooperative dynamic games for general insurance markets," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 123-135.
    12. Dutang, Christophe & Albrecher, Hansjoerg & Loisel, Stéphane, 2013. "Competition among non-life insurers under solvency constraints: A game-theoretic approach," European Journal of Operational Research, Elsevier, vol. 231(3), pages 702-711.
    13. Jeffrey R. Brown & Austan Goolsbee, 2002. "Does the Internet Make Markets More Competitive? Evidence from the Life Insurance Industry," Journal of Political Economy, University of Chicago Press, vol. 110(3), pages 481-507, June.
    14. Pantelous, Athanasios A. & Passalidou, Eudokia, 2013. "Optimal premium pricing policy in a competitive insurance market environment," Annals of Actuarial Science, Cambridge University Press, vol. 7(2), pages 175-191, September.
    15. Taksar, Michael & Zeng, Xudong, 2011. "Optimal non-proportional reinsurance control and stochastic differential games," Insurance: Mathematics and Economics, Elsevier, vol. 48(1), pages 64-71, January.
    16. Borch, Karl, 1962. "Application of Game Theory to Some Problems in Automobile Insurance*)," ASTIN Bulletin, Cambridge University Press, vol. 2(2), pages 208-221, September.
    17. Michael Powers & Martin Shubik & Shun Yao, 1998. "Insurance market games: Scale effects and public policy," Journal of Economics, Springer, vol. 67(2), pages 109-134, June.
    18. Xiang Lin & Chunhong Zhang & Tak Siu, 2012. "Stochastic differential portfolio games for an insurer in a jump-diffusion risk process," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 75(1), pages 83-100, February.
    19. Pun, Chi Seng & Wong, Hoi Ying, 2016. "Robust non-zero-sum stochastic differential reinsurance game," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 169-177.
    20. Wu, Renchao & Pantelous, Athanasios A., 2017. "Potential Games With Aggregation In Non-Cooperative General Insurance Markets," ASTIN Bulletin, Cambridge University Press, vol. 47(1), pages 269-302, January.
    21. Taylor, G. C., 1987. "Expenses and underwriting strategy in competition," Insurance: Mathematics and Economics, Elsevier, vol. 6(4), pages 275-287, November.
    22. Peter A. Diamond, 1982. "Wage Determination and Efficiency in Search Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 49(2), pages 217-227.
    23. Jean-Philippe Boucher & Michel Denuit & Montserrat Guillén, 2007. "Risk Classification for Claim Counts," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(4), pages 110-131.
    24. Mortensen, Dale T, 1982. "Property Rights and Efficiency in Mating, Racing, and Related Games," American Economic Review, American Economic Association, vol. 72(5), pages 968-979, December.
    25. Emms, Paul, 2007. "Dynamic Pricing of General Insurance in a Competitive Market," ASTIN Bulletin, Cambridge University Press, vol. 37(1), pages 1-34, May.
    26. Taylor, G. C., 1986. "Underwriting strategy in a competitive insurance environment," Insurance: Mathematics and Economics, Elsevier, vol. 5(1), pages 59-77, January.
    27. repec:dau:papers:123456789/1908 is not listed on IDEAS
    28. Yan, Ming & Peng, Fanyi & Zhang, Shuhua, 2017. "A reinsurance and investment game between two insurance companies with the different opinions about some extra information," Insurance: Mathematics and Economics, Elsevier, vol. 75(C), pages 58-70.
    29. Paul Emms, 2012. "Equilibrium Pricing of General Insurance Policies," North American Actuarial Journal, Taylor & Francis Journals, vol. 16(3), pages 323-349.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Bai, Yanfei & Zhou, Zhongbao & Xiao, Helu & Gao, Rui & Zhong, Feimin, 2022. "A hybrid stochastic differential reinsurance and investment game with bounded memory," European Journal of Operational Research, Elsevier, vol. 296(2), pages 717-737.
    2. Yanfei Bai & Zhongbao Zhou & Helu Xiao & Rui Gao & Feimin Zhong, 2019. "A hybrid stochastic differential reinsurance and investment game with bounded memory," Papers 1910.09834, arXiv.org.
    3. Yevhen Havrylenko & Maria Hinken & Rudi Zagst, 2022. "Risk sharing in equity-linked insurance products: Stackelberg equilibrium between an insurer and a reinsurer," Papers 2203.04053, arXiv.org, revised Oct 2023.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Asmussen, Søren & Christensen, Bent Jesper & Thøgersen, Julie, 2019. "Nash equilibrium premium strategies for push–pull competition in a frictional non-life insurance market," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 92-100.
    2. Boonen, Tim J. & Pantelous, Athanasios A. & Wu, Renchao, 2018. "Non-cooperative dynamic games for general insurance markets," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 123-135.
    3. Mourdoukoutas, Fotios & Boonen, Tim J. & Koo, Bonsoo & Pantelous, Athanasios A., 2021. "Pricing in a competitive stochastic insurance market," Insurance: Mathematics and Economics, Elsevier, vol. 97(C), pages 44-56.
    4. Claire Mouminoux & Christophe Dutang & Stéphane Loisel & Hansjoerg Albrecher, 2022. "On a Markovian Game Model for Competitive Insurance Pricing," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 1061-1091, June.
    5. Dutang, Christophe & Albrecher, Hansjoerg & Loisel, Stéphane, 2013. "Competition among non-life insurers under solvency constraints: A game-theoretic approach," European Journal of Operational Research, Elsevier, vol. 231(3), pages 702-711.
    6. Christensen, Bent Jesper & Parra-Alvarez, Juan Carlos & Serrano, Rafael, 2021. "Optimal control of investment, premium and deductible for a non-life insurance company," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 384-405.
    7. Pantelous, Athanasios A. & Passalidou, Eudokia, 2015. "Optimal premium pricing strategies for competitive general insurance markets," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 858-874.
    8. Zhu, Huainian & Cao, Ming & Zhang, Chengke, 2019. "Time-consistent investment and reinsurance strategies for mean-variance insurers with relative performance concerns under the Heston model," Finance Research Letters, Elsevier, vol. 30(C), pages 280-291.
    9. Hong Mao & Zhongkai Wen, 2020. "Optimal Decision on Dynamic Insurance Price and Investment Portfolio of an Insurer with Multi-dimensional Time-Varying Correlation," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 18(1), pages 29-51, March.
    10. Wang, Ning & Zhang, Nan & Jin, Zhuo & Qian, Linyi, 2021. "Stochastic differential investment and reinsurance games with nonlinear risk processes and VaR constraints," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 168-184.
    11. Mao, Hong & Carson, James M. & Ostaszewski, Krzysztof M. & Wen, Zhongkai, 2013. "Optimal decision on dynamic insurance price and investment portfolio of an insurer," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 359-369.
    12. Nobuhiro Kiyotaki & Ricardo Lagos, 2007. "A Model of Job and Worker Flows," Journal of Political Economy, University of Chicago Press, vol. 115(5), pages 770-819, October.
    13. Paolo Martellini & Guido Menzio, 2020. "Declining Search Frictions, Unemployment, and Growth," Journal of Political Economy, University of Chicago Press, vol. 128(12), pages 4387-4437.
    14. Coen N. Teulings & Pieter A. Gautier, 2004. "The Right Man for the Job," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 71(2), pages 553-580.
    15. Yashiv, Eran, 2007. "Labor search and matching in macroeconomics," European Economic Review, Elsevier, vol. 51(8), pages 1859-1895, November.
    16. Anja Bauer & Ian King, 2015. "The Hartz Reforms, the German Miracle, and the Reallocation Puzzle," Discussion Papers Series 550, School of Economics, University of Queensland, Australia.
    17. Yashiv, Eran, 2006. "Evaluating the performance of the search and matching model," European Economic Review, Elsevier, vol. 50(4), pages 909-936, May.
    18. Mr. Ruy Lama & Juan Pablo Medina Guzman, 2015. "Fiscal Consolidation During Times of High Unemployment: The Role of Productivity Gains and Wage Restraint," IMF Working Papers 2015/262, International Monetary Fund.
    19. Derek Stacey, 2019. "Posted Prices, Search and Bargaining," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 33, pages 85-104, July.
    20. Emms, P. & Haberman, S. & Savoulli, I., 2007. "Optimal strategies for pricing general insurance," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 15-34, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jrisks:v:7:y:2019:i:2:p:49-:d:227500. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.