
-
Previous Article
Impact of reorder option in supply chain coordination
- JIMO Home
- This Issue
-
Next Article
Multiple-stage multiple-machine capacitated lot-sizing and scheduling with sequence-dependent setup: A case study in the wheel industry
Dynamic pricing of network goods in duopoly markets with boundedly rational consumers
1. | Business School of Central South University, Changsha, Hunan 410083, China |
2. | School of Accountancy, Hunan University of Finance and Economics, Changsha, Hunan 410205, China |
3. | Department of Mathematics and Statistics, Curtin University, Australia |
In this paper, we present a dynamic pricing model for two firms selling products displaying network effects for which consumers are with bounded rationality. We formulate this model in the form of differential games and derive the open-loop equilibrium prices for the firms. Then, we show the existence and uniqueness of such open-loop equilibrium prices. The model is further extended to the case with heterogeneous network effects. Their steady-state prices obtained are compared. A numerical example is solved and the results obtained are used to analyze how the steady-state prices and market shares of both firms are influenced by the cost, price sensitivity and the network effects of the products.
References:
[1] |
S. P. Anderson and A. D. Palma,
Multiproduct firms: A nested logit approach, The Journal of Industrial Economics, 40 (1992), 261-276.
doi: 10.2307/2950539. |
[2] |
S. P. Anderson, A. D. Palma and J. F. Thisse,
Discrete Choice Theory of Product Differentiation, MIT press, 1992. |
[3] |
M. E. Ben-Akiva and S. R. Lerman, Discrete Choice Analysis: Theory and Application to Travel Demand, MIT press, 1985. Google Scholar |
[4] |
B. Bensaid and J. P. Lesne,
Dynamic monopoly pricing with network externalities, International Journal of Industrial Organization, 14 (1996), 837-855.
doi: 10.1016/0167-7187(95)01000-9. |
[5] |
F. Bloch and N. Quérou,
Pricing in social networks, Games and economic behavior, 80 (2013), 243-261.
doi: 10.1016/j.geb.2013.03.006. |
[6] |
L. Cabral,
Dynamic price competition with network effects, The Review of Economic Studies, 78 (2011), 83-111.
doi: 10.1093/restud/rdq007. |
[7] |
L. Cabral, D. J. Salant and G. A. Woroch, Monopoly pricing with network externalities, International Journal of Industrial Organization, 17 (1999), 199-214. Google Scholar |
[8] |
O. Candogan, K. Bimpikis and A. Ozdaglar,
Optimal pricing in networks with externalities, Operations Research, 60 (2012), 883-905.
doi: 10.1287/opre.1120.1066. |
[9] |
A. Caplin and B. Nalebuff,
Aggregation and imperfect competition: On the existence of equilibrium, The Econometric Society, 59 (1991), 25-59.
doi: 10.2307/2938239. |
[10] |
P. K. Chintagunta and V. R. Rao,
Pricing strategies in a dynamic duopoly: A differential game model, Management Science, 42 (1996), 1501-1514.
doi: 10.1287/mnsc.42.11.1501. |
[11] |
E. Damiano and L. Hao,
Competing matchmaking, Journal of the European Economic Association, 6 (2008), 789-818.
doi: 10.1162/JEEA.2008.6.4.789. |
[12] |
A. Dhebar and S. S. Oren,
Optimal dynamic pricing for expanding networks, Marketing Science, 4 (1985), 336-351.
doi: 10.1287/mksc.4.4.336. |
[13] |
T. Doganoglu, Dynamic price competition with consumption externalities, Netnomics, 5 (2003), 43-69. Google Scholar |
[14] |
C. Du, W. L. Cooper and Z. Wang, Optimal Pricing for a Multinomial Logit Choice Model with Network Effects, 2014. Available at SSRN 2477548: http://ssrn.com/abstract=2477548. Google Scholar |
[15] |
N. Economides, The economics of networks, International journal of industrial organization, 14 (1996), 673-699. Google Scholar |
[16] |
G. Ellison and D. Fudenberg, Knife-edge or plateau: When do market models tip?, The Quarterly Journal of Economics, 118 (2003), 1249-1278. Google Scholar |
[17] |
G. Ellison, D. Fudenberg and M. Möbius, Competing auctions, Journal of the European Economic Association, 2 (2004), 30-66. Google Scholar |
[18] |
W. Elmaghraby and P. Keskinocak, Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions, Management Science, 49 (2003), 1287-1309. Google Scholar |
[19] |
J. Farrell and G. Saloner, Standardization, compatibility, and innovation, The RAND Journal of Economics, 16 (1985), 70-83. Google Scholar |
[20] |
X. Gabaix, A sparsity-based model of bounded rationality, The Quarterly Journal of Economics, 129 (2014), 1661-1710. Google Scholar |
[21] |
X. Gabaix, Sparse dynamic programming and aggregate fluctuations, manuscript, 2013. Google Scholar |
[22] |
A. Herbon,
Dynamic pricing vs. acquiring information on consumers' heterogeneous sensitivity to product freshness, International Journal of Production Research, 52 (2014), 918-933.
doi: 10.1080/00207543.2013.843800. |
[23] |
E. Hopkins,
Adaptive learning models of consumer behavior, Journal of economic behavior & organization, 64 (2007), 348-368.
doi: 10.1016/j.jebo.2006.02.010. |
[24] |
M. L. Katz and C. Shapiro, Network externalities, competition, and compatibility, The American economic review, 75 (2014), 424-440. Google Scholar |
[25] |
M. J. Khouja,
Optimal ordering, discounting, and pricing in the single-period problem, International Journal of Production Economics, 65 (2000), 201-216.
doi: 10.1016/S0925-5273(99)00027-4. |
[26] |
D. Laussel and J. Resende,
Dynamic price competition in aftermarkets with network effects, Journal of Mathematical Economics, 50 (2014), 106-118.
doi: 10.1016/j.jmateco.2013.10.002. |
[27] |
Y. Levin, J. McGill and M. Nediak,
Dynamic pricing in the presence of strategic consumers and oligopolistic competition, Management Science, 55 (2008), 32-46.
doi: 10.1287/mnsc.1080.0936. |
[28] |
M. F. Mitchell and A. Skrzypacz,
Network externalities and long-run market shares, Economic Theory, 29 (2006), 621-648.
doi: 10.1007/s00199-005-0031-0. |
[29] |
R. Radner, A. Radunskaya and A. Sundararajan,
Dynamic pricing of network goods with boundedly rational consumers, Proceedings of the National Academy of Sciences, 111 (2014), 99-104.
doi: 10.1073/pnas.1319543110. |
[30] |
X. Vives, Oligopoly Pricing: Old Ideas and New Tools, MIT press, 2001. Google Scholar |
[31] |
X. L. Xu and X. Q. Cai,
Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium, Journal of Industrial and Management Optimization, 4 (2008), 843-859.
doi: 10.3934/jimo.2008.4.843. |
[32] |
J. X. Zhang, Z. Y. Bai and W. S. Tang,
Optimal pricing policy for deteriorating items with preservation technology investment, Journal of Industrial and Management Optimization, 10 (2014), 1261-1277.
doi: 10.3934/jimo.2014.10.1261. |
show all references
References:
[1] |
S. P. Anderson and A. D. Palma,
Multiproduct firms: A nested logit approach, The Journal of Industrial Economics, 40 (1992), 261-276.
doi: 10.2307/2950539. |
[2] |
S. P. Anderson, A. D. Palma and J. F. Thisse,
Discrete Choice Theory of Product Differentiation, MIT press, 1992. |
[3] |
M. E. Ben-Akiva and S. R. Lerman, Discrete Choice Analysis: Theory and Application to Travel Demand, MIT press, 1985. Google Scholar |
[4] |
B. Bensaid and J. P. Lesne,
Dynamic monopoly pricing with network externalities, International Journal of Industrial Organization, 14 (1996), 837-855.
doi: 10.1016/0167-7187(95)01000-9. |
[5] |
F. Bloch and N. Quérou,
Pricing in social networks, Games and economic behavior, 80 (2013), 243-261.
doi: 10.1016/j.geb.2013.03.006. |
[6] |
L. Cabral,
Dynamic price competition with network effects, The Review of Economic Studies, 78 (2011), 83-111.
doi: 10.1093/restud/rdq007. |
[7] |
L. Cabral, D. J. Salant and G. A. Woroch, Monopoly pricing with network externalities, International Journal of Industrial Organization, 17 (1999), 199-214. Google Scholar |
[8] |
O. Candogan, K. Bimpikis and A. Ozdaglar,
Optimal pricing in networks with externalities, Operations Research, 60 (2012), 883-905.
doi: 10.1287/opre.1120.1066. |
[9] |
A. Caplin and B. Nalebuff,
Aggregation and imperfect competition: On the existence of equilibrium, The Econometric Society, 59 (1991), 25-59.
doi: 10.2307/2938239. |
[10] |
P. K. Chintagunta and V. R. Rao,
Pricing strategies in a dynamic duopoly: A differential game model, Management Science, 42 (1996), 1501-1514.
doi: 10.1287/mnsc.42.11.1501. |
[11] |
E. Damiano and L. Hao,
Competing matchmaking, Journal of the European Economic Association, 6 (2008), 789-818.
doi: 10.1162/JEEA.2008.6.4.789. |
[12] |
A. Dhebar and S. S. Oren,
Optimal dynamic pricing for expanding networks, Marketing Science, 4 (1985), 336-351.
doi: 10.1287/mksc.4.4.336. |
[13] |
T. Doganoglu, Dynamic price competition with consumption externalities, Netnomics, 5 (2003), 43-69. Google Scholar |
[14] |
C. Du, W. L. Cooper and Z. Wang, Optimal Pricing for a Multinomial Logit Choice Model with Network Effects, 2014. Available at SSRN 2477548: http://ssrn.com/abstract=2477548. Google Scholar |
[15] |
N. Economides, The economics of networks, International journal of industrial organization, 14 (1996), 673-699. Google Scholar |
[16] |
G. Ellison and D. Fudenberg, Knife-edge or plateau: When do market models tip?, The Quarterly Journal of Economics, 118 (2003), 1249-1278. Google Scholar |
[17] |
G. Ellison, D. Fudenberg and M. Möbius, Competing auctions, Journal of the European Economic Association, 2 (2004), 30-66. Google Scholar |
[18] |
W. Elmaghraby and P. Keskinocak, Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions, Management Science, 49 (2003), 1287-1309. Google Scholar |
[19] |
J. Farrell and G. Saloner, Standardization, compatibility, and innovation, The RAND Journal of Economics, 16 (1985), 70-83. Google Scholar |
[20] |
X. Gabaix, A sparsity-based model of bounded rationality, The Quarterly Journal of Economics, 129 (2014), 1661-1710. Google Scholar |
[21] |
X. Gabaix, Sparse dynamic programming and aggregate fluctuations, manuscript, 2013. Google Scholar |
[22] |
A. Herbon,
Dynamic pricing vs. acquiring information on consumers' heterogeneous sensitivity to product freshness, International Journal of Production Research, 52 (2014), 918-933.
doi: 10.1080/00207543.2013.843800. |
[23] |
E. Hopkins,
Adaptive learning models of consumer behavior, Journal of economic behavior & organization, 64 (2007), 348-368.
doi: 10.1016/j.jebo.2006.02.010. |
[24] |
M. L. Katz and C. Shapiro, Network externalities, competition, and compatibility, The American economic review, 75 (2014), 424-440. Google Scholar |
[25] |
M. J. Khouja,
Optimal ordering, discounting, and pricing in the single-period problem, International Journal of Production Economics, 65 (2000), 201-216.
doi: 10.1016/S0925-5273(99)00027-4. |
[26] |
D. Laussel and J. Resende,
Dynamic price competition in aftermarkets with network effects, Journal of Mathematical Economics, 50 (2014), 106-118.
doi: 10.1016/j.jmateco.2013.10.002. |
[27] |
Y. Levin, J. McGill and M. Nediak,
Dynamic pricing in the presence of strategic consumers and oligopolistic competition, Management Science, 55 (2008), 32-46.
doi: 10.1287/mnsc.1080.0936. |
[28] |
M. F. Mitchell and A. Skrzypacz,
Network externalities and long-run market shares, Economic Theory, 29 (2006), 621-648.
doi: 10.1007/s00199-005-0031-0. |
[29] |
R. Radner, A. Radunskaya and A. Sundararajan,
Dynamic pricing of network goods with boundedly rational consumers, Proceedings of the National Academy of Sciences, 111 (2014), 99-104.
doi: 10.1073/pnas.1319543110. |
[30] |
X. Vives, Oligopoly Pricing: Old Ideas and New Tools, MIT press, 2001. Google Scholar |
[31] |
X. L. Xu and X. Q. Cai,
Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium, Journal of Industrial and Management Optimization, 4 (2008), 843-859.
doi: 10.3934/jimo.2008.4.843. |
[32] |
J. X. Zhang, Z. Y. Bai and W. S. Tang,
Optimal pricing policy for deteriorating items with preservation technology investment, Journal of Industrial and Management Optimization, 10 (2014), 1261-1277.
doi: 10.3934/jimo.2014.10.1261. |
the price of Firm |
|
the probability of a consumer purchases Firm |
|
consumer's time-varying preference for Firm |
|
the coefficient of the time-varying preference for Firm |
|
the price sensitivity parameter for Firm |
|
market's total demand of Firm |
|
the network effects sensitivity parameter for Firm |
|
the rate of change of |
|
the stochastic utility gained by the consumer for purchasing Firm |
|
the fraction of consumers in segment |
|
the probability of purchasing Firm |
|
the time varying preference of Firm |
the price of Firm |
|
the probability of a consumer purchases Firm |
|
consumer's time-varying preference for Firm |
|
the coefficient of the time-varying preference for Firm |
|
the price sensitivity parameter for Firm |
|
market's total demand of Firm |
|
the network effects sensitivity parameter for Firm |
|
the rate of change of |
|
the stochastic utility gained by the consumer for purchasing Firm |
|
the fraction of consumers in segment |
|
the probability of purchasing Firm |
|
the time varying preference of Firm |
[1] |
Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 |
[2] |
Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 |
[3] |
Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 |
[4] |
Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53 |
[5] |
Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 |
[6] |
Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 |
[7] |
Xianchao Xiu, Ying Yang, Wanquan Liu, Lingchen Kong, Meijuan Shang. An improved total variation regularized RPCA for moving object detection with dynamic background. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1685-1698. doi: 10.3934/jimo.2019024 |
[8] |
Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021035 |
[9] |
Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243 |
[10] |
Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 |
[11] |
Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637 |
[12] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[13] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[14] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
[15] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]