Dynamic pricing of network goods in duopoly markets with boundedly rational consumers

Haiying Liu; Xinxing Luo; Wenjie Bi; Yueming Man; Kok Lay Teo

doi:10.3934/jimo.2016025

Article Contents

2017, Volume 13, Issue 1: 429-447. Doi: 10.3934/jimo.2016025

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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
¹ Corresponding Author Email:beenjoy@126.com

Received: March 31, 2015

Published: August 2017

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 91A80; Secondary: 91A23.

Citation:

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Figure 1. The relationship between $p^*_i$ and $\alpha_1$ when $ \alpha_2=0.95 $

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Figure 2. The relationship between $q^*_i$ and $\alpha_1$ when $ \alpha_2=0.95 $

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Figure 3. The relationship between $p^*_i$ and $c_1$ when $ c_2=1 $

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Figure 4. The relationship between $q^*_i$ and $c_1$ when $ c_2=1 $

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Figure 5. The relationship between $p^*_i$ and $\beta_1$ when $ \beta_2=0.9 $

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Figure 6. The relationship between $q^*_i$ and $\beta_1$ when $ \beta_2=0.9 $

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Figure 7. The relationship between $p^*_i$ with $\gamma_1$ when $ \gamma_2=0.8 $

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Figure 8. The relationship between $q^*_i$ with $\gamma_1$ when $ \gamma_2=0.8 $

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Figure 9. The relationship between $J^*_i$ and $\alpha_1$ when $ \alpha_2=0.95 $

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Figure 10. The relationship between $J^*_i$ and $c_1$ when $ c_2=1 $

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Figure 11. The relationship between $J^*_i$ and $\beta_1$ when $ \beta_2=0.9 $

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Figure 12. The relationship between $J^*_i$ and $\gamma_1$ when $ \gamma_2=0.8 $

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Figure 13. Comparison between the $p^*_1$ and $p^m_1$

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Figure 14. Comparison between the $p^*_i$ and $p^W_i$

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$ 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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