# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022056
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## A dynamic analysis of a monopolist's product and process innovation with nonlinear demand and expected quality effects

 Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200052, China

Received  January 2022 Revised  March 2022 Early access April 2022

The purpose of this paper is to investigate the dynamic control problems of a firm's product innovation (quality-improving) and process innovation (cost-reducing) with expected quality (reference quality) effects under nonlinear demand in a monopoly market. Our work significant features are: (i) a monopolist dealing with customer behavior in the spirit of behavioral economics determines the product quality, and carries out product and process innovation activities over time; (ii) the consumers' demand structure depends on product quality, expected quality and price in a separable multiplicative way between state variables and control variable. Our main results show (i) under monopolist decision-making and social planner adjustment, the stability of the system depends on the discount rate and consumers' memory parameter; (ii) the effort rate of process innovation is increasing with the expected quality, while the effort rate of product innovation is increasing with the memory parameter in the neighborhood of the steady-state shadow price of expected quality; (iii) as the memory parameter increases, the steady-state effort of process innovation is greater than that of product innovation; (iv) although the price is still determined by the monopolist under social planner adjustment, the price as well as both the efforts of product and process innovation under the social planner adjustment are always higher than that under the monopolist decision-making.

Citation: Shoude Li. A dynamic analysis of a monopolist's product and process innovation with nonlinear demand and expected quality effects. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022056
##### References:
 [1] R. Cellini and L. Lambertini, Dynamic R & D with spillovers: Competition vs cooperation, Journal of Economic Dynamics & Control, 33 (2009), 568-582.  doi: 10.1016/j.jedc.2008.08.006. [2] Y. M. Chen and M. Schwartz, Product innovation incentives: monopoly vs. competition, Journal of Economics & Management Strategy, 22(3) (2013), 513-528. [3] H. Dawid, Y. K. Michel, K. Michael, M. Peter and P. M. Kort, Product innovation incentives by an incumbent firm: A dynamic analysis, Journal of Economic Behavior & Organization, 117 (2015), 411-438.  doi: 10.1016/j.jet.2015.01.001. [4] W. Kim and and M. Kim, Reference quality-based competitive market structure for innovation driven markets, International Journal of Research in Marketing, 32 (2015), 284-296.  doi: 10.1016/j.ijresmar.2014.10.003. [5] D. Kahnemann and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47(2) (1979), 263-291. doi: 10.2307/1914185. [6] P. K. Kopalle and R. S. Winer, A dynamic model of reference price and expected quality, Marketing Letters, 7 (1) (1996), 41-52. doi: 10.1007/BF00557310. [7] L. Lambertini, The monopolist's optimal R & D portfolio, Oxford Economic Papers, 55 (2003), 561-578. doi: 10.1093/oep/55.4.561. [8] L. Lambertini, Process and product R & D by a multiproduct monopolist: a reply to Lin., Oxford Economic Papers, 56 (2004), 745-749. doi: 10.2307/3488808. [9] L. Lambertini and A. Mantovani, Process and product innovation by a multiproduct monopolist: A dynamic approach, International Journal of Industrial Organization, 27(4) (2009), 508-518.  doi: 10.1016/j.ijindorg.2008.12.005. [10] L. Lambertini and A. Mantovani, Process and product innovation: A differential game approach to product life cycle, International Journal of Economic Theory, 6 (2010), 227-252.  doi: 10.1111/j.1742-7363.2010.00132.x. [11] L. Lambertini and R. Orsini, Quality improvement and process innovation in monopoly: A dynamic analysis, Operations Research Letters, 43 (2015), 370-373.  doi: 10.1016/j.orl.2015.04.009. [12] L. Lambertini, R. Orsini and A. Palestini, On the instability of the R & D portfolio in a dynamic monopoly. Or, one cannot get two eggs in one basket, International Journal of Production Economics, 193 (2017), 703-712. doi: 10.1016/j.ijpe.2017.08.030. [13] S. D. Li and J. Ni, A dynamic analysis of investment in process and product innovation with learning by doing, Economics Letters, 145 (2016), 104-108. doi: 10.1016/j.econlet.2016.05.031. [14] S. D. Li, Dynamic control of a multiproduct monopolist firm's product and process innovation, Journal of the Operational Research Society, 69(5) (2018), 714-733. doi: 10.1057/s41274-017-0260-1. [15] Z. Li and J. Ni, Dynamic product innovation and production decisions under quality authorization, Computers & Industrial Engineering, 116 (2018), 13-21. doi: 10.1016/j.cie.2017.12.011. [16] P. Lin, Process and product R & D by a multiproduct monopolist, Oxford Economic Papers, 56 (2004), 735-743. doi: 10.1093/oep/gpf065. [17] A. Mantovani, Complementarity between product and process innovation in a monopoly setting, Economics of Innovation and New Technology, 15 (3) (2006), 219-234. doi: 10.6092/unibo. [18] G. Martín-Herrán, S. Taboubi and G. Zaccour, Dual role of price and myopia in a marketing channel, Economics of Innovation and New Technology, 219(2) (2006), 284-295. doi: 10.1016/j.ejor.2011.12.015. [19] F. El Ouardighi and K. Kogan, Dynamic conformance and design quality in a supply chain: an assessment of contracts' coordinating power, Annals of Operations Research, 211 (2013), 137-166. doi: 10.1007/s10479-013-1414-4. [20] F. El Ouardighi, Supply quality management with optimal wholesale price and revenue sharing contracts: a two-stage game approach, International Journal of Production Economics, 156 (2014), 260-268. doi: 10.1016/j.ijpe.2014.06.006. [21] X. J. Pan and S. D. Li, Dynamic optimal control of process-product innovation with learning by doing, European Journal of Operational Research, 248 (2016), 136-145. doi: 10.1016/j.ejor.2015.07.007. [22] S. Rosenkranz, Simultaneous choice of process and product innovation, Journal of Economic Behavior & Organization, 50(2) (2003), 183-201. [23] G. Y. Zhong and W. H. Zhang, Product and process innovation with knowledge accumulation in monopoly: A dynamic analysis, Economics Letters, 163 (2018), 175-178. doi: 10.1016/j.econlet.2017.12.016.

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##### References:
 [1] R. Cellini and L. Lambertini, Dynamic R & D with spillovers: Competition vs cooperation, Journal of Economic Dynamics & Control, 33 (2009), 568-582.  doi: 10.1016/j.jedc.2008.08.006. [2] Y. M. Chen and M. Schwartz, Product innovation incentives: monopoly vs. competition, Journal of Economics & Management Strategy, 22(3) (2013), 513-528. [3] H. Dawid, Y. K. Michel, K. Michael, M. Peter and P. M. Kort, Product innovation incentives by an incumbent firm: A dynamic analysis, Journal of Economic Behavior & Organization, 117 (2015), 411-438.  doi: 10.1016/j.jet.2015.01.001. [4] W. Kim and and M. Kim, Reference quality-based competitive market structure for innovation driven markets, International Journal of Research in Marketing, 32 (2015), 284-296.  doi: 10.1016/j.ijresmar.2014.10.003. [5] D. Kahnemann and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47(2) (1979), 263-291. doi: 10.2307/1914185. [6] P. K. Kopalle and R. S. Winer, A dynamic model of reference price and expected quality, Marketing Letters, 7 (1) (1996), 41-52. doi: 10.1007/BF00557310. [7] L. Lambertini, The monopolist's optimal R & D portfolio, Oxford Economic Papers, 55 (2003), 561-578. doi: 10.1093/oep/55.4.561. [8] L. Lambertini, Process and product R & D by a multiproduct monopolist: a reply to Lin., Oxford Economic Papers, 56 (2004), 745-749. doi: 10.2307/3488808. [9] L. Lambertini and A. Mantovani, Process and product innovation by a multiproduct monopolist: A dynamic approach, International Journal of Industrial Organization, 27(4) (2009), 508-518.  doi: 10.1016/j.ijindorg.2008.12.005. [10] L. Lambertini and A. Mantovani, Process and product innovation: A differential game approach to product life cycle, International Journal of Economic Theory, 6 (2010), 227-252.  doi: 10.1111/j.1742-7363.2010.00132.x. [11] L. Lambertini and R. Orsini, Quality improvement and process innovation in monopoly: A dynamic analysis, Operations Research Letters, 43 (2015), 370-373.  doi: 10.1016/j.orl.2015.04.009. [12] L. Lambertini, R. Orsini and A. Palestini, On the instability of the R & D portfolio in a dynamic monopoly. Or, one cannot get two eggs in one basket, International Journal of Production Economics, 193 (2017), 703-712. doi: 10.1016/j.ijpe.2017.08.030. [13] S. D. Li and J. Ni, A dynamic analysis of investment in process and product innovation with learning by doing, Economics Letters, 145 (2016), 104-108. doi: 10.1016/j.econlet.2016.05.031. [14] S. D. Li, Dynamic control of a multiproduct monopolist firm's product and process innovation, Journal of the Operational Research Society, 69(5) (2018), 714-733. doi: 10.1057/s41274-017-0260-1. [15] Z. Li and J. Ni, Dynamic product innovation and production decisions under quality authorization, Computers & Industrial Engineering, 116 (2018), 13-21. doi: 10.1016/j.cie.2017.12.011. [16] P. Lin, Process and product R & D by a multiproduct monopolist, Oxford Economic Papers, 56 (2004), 735-743. doi: 10.1093/oep/gpf065. [17] A. Mantovani, Complementarity between product and process innovation in a monopoly setting, Economics of Innovation and New Technology, 15 (3) (2006), 219-234. doi: 10.6092/unibo. [18] G. Martín-Herrán, S. Taboubi and G. Zaccour, Dual role of price and myopia in a marketing channel, Economics of Innovation and New Technology, 219(2) (2006), 284-295. doi: 10.1016/j.ejor.2011.12.015. [19] F. El Ouardighi and K. Kogan, Dynamic conformance and design quality in a supply chain: an assessment of contracts' coordinating power, Annals of Operations Research, 211 (2013), 137-166. doi: 10.1007/s10479-013-1414-4. [20] F. El Ouardighi, Supply quality management with optimal wholesale price and revenue sharing contracts: a two-stage game approach, International Journal of Production Economics, 156 (2014), 260-268. doi: 10.1016/j.ijpe.2014.06.006. [21] X. J. Pan and S. D. Li, Dynamic optimal control of process-product innovation with learning by doing, European Journal of Operational Research, 248 (2016), 136-145. doi: 10.1016/j.ejor.2015.07.007. [22] S. Rosenkranz, Simultaneous choice of process and product innovation, Journal of Economic Behavior & Organization, 50(2) (2003), 183-201. [23] G. Y. Zhong and W. H. Zhang, Product and process innovation with knowledge accumulation in monopoly: A dynamic analysis, Economics Letters, 163 (2018), 175-178. doi: 10.1016/j.econlet.2017.12.016.
The paths of the product innovation efforts $\hat{k}(t)$ and $\hat{k}(t)$ against time $t$
The paths of the product innovation efforts $\hat{h}(t)$ and $\hat{h}(t)$ against time $t$
The paths of the product innovation efforts $\hat{p}(t)$ and $\hat{p}(t)$ against time $t$
The parameters used in the numerical calculations
 $\rho$ $\delta$ $\mu$ $\alpha$ $\theta$ $\nu$ $\sigma$ $a_1$ $a_2$ $a_3$ $a$ 0.2 0.2 0.14 0.16 0.09 0.2 0.1 0.2 0.21 0.19 10
 $\rho$ $\delta$ $\mu$ $\alpha$ $\theta$ $\nu$ $\sigma$ $a_1$ $a_2$ $a_3$ $a$ 0.2 0.2 0.14 0.16 0.09 0.2 0.1 0.2 0.21 0.19 10
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