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January  2017, 13(1): 429-447. doi: 10.3934/jimo.2016025

## 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

1 Corresponding Author Email:beenjoy@126.com

Received  April 2015 Published  January 2016

Fund Project: This work is supported by NSFC NO.91646115?71371191,71210003,712221061, PSFC NO.2015JJ2194 and 2015CX010

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.

Citation: Haiying Liu, Xinxing Luo, Wenjie Bi, Yueming Man, Kok Lay Teo. Dynamic pricing of network goods in duopoly markets with boundedly rational consumers. Journal of Industrial & Management Optimization, 2017, 13 (1) : 429-447. doi: 10.3934/jimo.2016025
##### 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. [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. [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. [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. [15] N. Economides, The economics of networks, International journal of industrial organization, 14 (1996), 673-699. [16] G. Ellison and D. Fudenberg, Knife-edge or plateau: When do market models tip?, The Quarterly Journal of Economics, 118 (2003), 1249-1278. [17] G. Ellison, D. Fudenberg and M. Möbius, Competing auctions, Journal of the European Economic Association, 2 (2004), 30-66. [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. [19] J. Farrell and G. Saloner, Standardization, compatibility, and innovation, The RAND Journal of Economics, 16 (1985), 70-83. [20] X. Gabaix, A sparsity-based model of bounded rationality, The Quarterly Journal of Economics, 129 (2014), 1661-1710. [21] X. Gabaix, Sparse dynamic programming and aggregate fluctuations, manuscript, 2013. [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. [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. [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. [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. [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. [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. [15] N. Economides, The economics of networks, International journal of industrial organization, 14 (1996), 673-699. [16] G. Ellison and D. Fudenberg, Knife-edge or plateau: When do market models tip?, The Quarterly Journal of Economics, 118 (2003), 1249-1278. [17] G. Ellison, D. Fudenberg and M. Möbius, Competing auctions, Journal of the European Economic Association, 2 (2004), 30-66. [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. [19] J. Farrell and G. Saloner, Standardization, compatibility, and innovation, The RAND Journal of Economics, 16 (1985), 70-83. [20] X. Gabaix, A sparsity-based model of bounded rationality, The Quarterly Journal of Economics, 129 (2014), 1661-1710. [21] X. Gabaix, Sparse dynamic programming and aggregate fluctuations, manuscript, 2013. [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. [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. [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 relationship between $p^*_i$ and $\alpha_1$ when $\alpha_2=0.95$
The relationship between $q^*_i$ and $\alpha_1$ when $\alpha_2=0.95$
The relationship between $p^*_i$ and $c_1$ when $c_2=1$
The relationship between $q^*_i$ and $c_1$ when $c_2=1$
The relationship between $p^*_i$ and $\beta_1$ when $\beta_2=0.9$
The relationship between $q^*_i$ and $\beta_1$ when $\beta_2=0.9$
The relationship between $p^*_i$ with $\gamma_1$ when $\gamma_2=0.8$
The relationship between $q^*_i$ with $\gamma_1$ when $\gamma_2=0.8$
The relationship between $J^*_i$ and $\alpha_1$ when $\alpha_2=0.95$
The relationship between $J^*_i$ and $c_1$ when $c_2=1$
The relationship between $J^*_i$ and $\beta_1$ when $\beta_2=0.9$
The relationship between $J^*_i$ and $\gamma_1$ when $\gamma_2=0.8$
Comparison between the $p^*_1$ and $p^m_1$
Comparison between the $p^*_i$ and $p^W_i$
 $p_i(t)$ the price of Firm $i's$ product at time $t$ $q_i(t)$ the probability of a consumer purchases Firm $i's$ product $\xi_i(t)$ consumer's time-varying preference for Firm $i's$ product $\beta_i$ the coefficient of the time-varying preference for Firm $i's$ product $\gamma_i$ the price sensitivity parameter for Firm $i's$ product $x_i(t)$ market's total demand of Firm $i's$ product at time $t$ $\alpha_i$ the network effects sensitivity parameter for Firm $i's$ product $\dot{\xi}_{is}(t)$ the rate of change of $\xi_{i}(t)$ with respect to time $t$ $\varepsilon_i(t)$ the stochastic utility gained by the consumer for purchasing Firm $i's$ product at time $t$ $\varPsi_s$ the fraction of consumers in segment $s$ $q_{is}(t)$ the probability of purchasing Firm $i's$ product for consumers in segment $s$ $\xi_{is}(t)$ the time varying preference of Firm $i's$ product for consumers in segment $s$
 $p_i(t)$ the price of Firm $i's$ product at time $t$ $q_i(t)$ the probability of a consumer purchases Firm $i's$ product $\xi_i(t)$ consumer's time-varying preference for Firm $i's$ product $\beta_i$ the coefficient of the time-varying preference for Firm $i's$ product $\gamma_i$ the price sensitivity parameter for Firm $i's$ product $x_i(t)$ market's total demand of Firm $i's$ product at time $t$ $\alpha_i$ the network effects sensitivity parameter for Firm $i's$ product $\dot{\xi}_{is}(t)$ the rate of change of $\xi_{i}(t)$ with respect to time $t$ $\varepsilon_i(t)$ the stochastic utility gained by the consumer for purchasing Firm $i's$ product at time $t$ $\varPsi_s$ the fraction of consumers in segment $s$ $q_{is}(t)$ the probability of purchasing Firm $i's$ product for consumers in segment $s$ $\xi_{is}(t)$ the time varying preference of Firm $i's$ product for consumers in segment $s$
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