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doi: 10.3934/jimo.2022014
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## A dynamic analysis of a monopolist's quality improvement, process innovation and goodwill

 1 School of Economics and Management, Shanxi Normal University, Taiyuan 030002, China 2 Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200052, China

* Corresponding author: Shoude Li

Received  January 2020 Revised  September 2021 Early access February 2022

Although there are many literatures on firms' product and process innovation in recent years, the effects of advertising-based goodwill and consumers' reference quality on firms' product and process innovation are rarely considered. Therefore, the significant features of our study are: (i) introducing the concept and definition of the consumers' reference quality; (ii) considering the effect of product quality on goodwill; (iii) the customers' demand function depends on product quality, product price, advertising-based goodwill and the difference between product quality and reference quality, and the demand function takes a linear form. Our results suggest that (i) the system admits unique saddle-point steady-state equilibrium under the monopolist decision-making and the social planner regulation; (ii) with the increases of the product quality and goodwill, the corresponding investments also increases; however, with the increase of marginal cost and reference quality, the corresponding investment decreases; (iii) the monopolist's investment in one direction boosts the other in the neighborhood of the steady-state investments in product innovation, process innovation and advertising-based goodwill, respectively; and (iv) the monopolist will have an underinvestment problem as compared with the social planner.

Citation: Genlong Guo, Shoude Li. A dynamic analysis of a monopolist's quality improvement, process innovation and goodwill. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022014
##### 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] R. Chenavaz, Dynamic pricing, product and process innovation, European Journal of Operational Research, 222 (2012), 553-557.  doi: 10.1016/j.ejor.2012.05.009. [3] K. Choi, R. Narasimhanb and S. W. Kim, Opening the technological innovation black box: The case of the electronics industry in Korea, European Journal of Operational Research, 250 (2016), 192-203.  doi: 10.1016/j.ejor.2015.08.054. [4] G. M. Erickson, Dynamic Models of Advertising Competition, Second Edition, Dordrecht, Kluwer, Part of the International Series in Quantitative Marketing book series, 13, 2003. doi: 10.1007/978-1-4615-1031-4. [5] G. M. Erickson, A differential game model of the marketing-operations interface, European Journal of Operational Research, 211 (2011), 394-402.  doi: 10.1016/j.ejor.2010.11.016. [6] G. Feichtinger, R. Hartl and S. Sethi, Dynamic optimal control models in advertising: Recent developments, Management Science, 40 (1994), 195-226. [7] G. E. Fruchter, Signaling quality: Dynamic price-advertising model, Journal of Optimization Theory and Applications, 143 (2009), 479-496.  doi: 10.1007/s10957-009-9575-7. [8] L. Flach and M. Irlacher, Product versus process: Innovation strategies of multiproduct firms, American Economic Journal: Microeconomics, 10 (2018), 236-277.  doi: 10.1257/mic.20150272. [9] P. De Giovanni, Quality improvement vs. advertising support: Which strategy works better for a manufacturer?, European Journal of Operational Research, 208 (2011), 119-130.  doi: 10.1016/j.ejor.2010.08.003. [10] A. Gavious and O. Lowengart, Price-quality relationship in the presence of asymmetric dynamic reference quality effects, Marketing Letters, 23 (2012), 137-161.  doi: 10.1007/s11002-011-9143-4. [11] L. Grosset and B. Viscolani, Optimal dynamic advertising with an adverse exogenous effect on brand goodwill, Automatica, 45 (2009), 863-870. [12] S. Jørgensen and G. Zaccour, Differential Games in Marketing, Dordrecht, Kluwer, Springer, Boston, MA, 2004. doi: 10.1007/978-1-4419-8929-1. [13] D. Kahnemann and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-292.  doi: 10.2307/1914185. [14] P. K. Kopalle and R. S. Winer, A dynamic model of reference price and expected quality, Marketing Letters, 7 (1996), 41-52.  doi: 10.1007/BF00557310. [15] L. Lambertini, Differential Games in Industrial Economics, Cambridge, Cambridge University Press, 2018.  doi: 10.1017/9781316691175. [16] L. Lambertini, Advertising in a dynamic spatial monopoly, European Journal of Operational Research, 166 (2005), 547-556.  doi: 10.1016/j.ejor.2004.03.013. [17] L. Lambertini and A. Mantovani, Process and product innovation by a multiproduct monopolist: A dynamic approach, International Journal of Industrial Organization, 27 (2009), 508-518.  doi: 10.1016/j.ijindorg.2008.12.005. [18] 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. [19] 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. [20] 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. [21] L. Lambertini and A. Palestini, Dynamic advertising with spillovers: Cartel vs competitive fringe, Optimal Control Applications and Methods, 30 (2009), 562-572.  doi: 10.1002/oca.881. [22] S. D. Li, Dynamic control of a multiproduct monopolist firm's product and process innovation, Journal of the Operational Research Society, 69 (2018), 714-733.  doi: 10.1057/s41274-017-0260-1. [23] P. Lin and K. Saggi, Product differentiation, process R & D, and the nature of market competition, European Economic Review, 46 (2002), 201-211.  doi: 10.1016/S0014-2921(00)00090-8. [24] G. Liu, S. P. Sethi and J. Zhang, Myopic vs far-sighted behaviours in a revenue-sharing supply chain with reference quality effects, International Journal of Production Research, 54 (2016), 1334-1357.  doi: 10.1080/00207543.2015.1068962. [25] G. W. Liu, J. X. Zhang and W. S. Tang, Strategic transfer pricing in a marketing-operations interface with quality level and advertising dependent goodwill, Omega, 56 (2015), 1-15.  doi: 10.1016/j.omega.2015.01.004. [26] A. Mantovani, Complementarity between product and process innovation in a monopoly setting, Economics of Innovation and New Technology, 15 (2006), 219-234. [27] A. Nair and R. Narasimhan, Dynamics of competing with quality-and advertising-based goodwill, European Journal of Operational Research, 175 (2006), 462-474.  doi: 10.1016/j.ejor.2005.05.015. [28] M. Nerlove and K. J. Arrow, Optimal advertising policy under dynamic conditions, Mathematical Models in Marketing, 132 (1976), 167-168.  doi: 10.1007/978-3-642-51565-1_54. [29] F. E. Ouardighi and C. S. Tapiero, Quality and the diffusion of innovations, European Journal of Operational Research, 106 (1998), 31-38.  doi: 10.1016/S0377-2217(97)00158-6. [30] F. E. Ouardighi and F. Pasin, Quality improvement and goodwill accumulation in a dynamic duopoly, European Journal of Operational Research, 175 (2006), 1021-1032.  doi: 10.1016/j.ejor.2005.06.020. [31] 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. [32] I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Operations Research, 55 (2007), 413-429. [33] S. Saha, Consumer preferences and product and process R & D, The RAND Journal of Economics, 38 (2007), 250–268. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.537.5906&rep=rep1&type=pdf. [34] G. Sorger, Reference price formation and optimal marketing strategies, Optimal control theory and economic analysis, 3 (1988), 97-120. [35] S. P. Sethi, Dynamic optimal control models in advertising: A survey, SIAM Review, 19 (1977), 685-725.  doi: 10.1137/1019106. [36] A. M. Spence, Monopoly, quality and regulation, Bell Journal of Economics, 6 (1975), 417-429.  doi: 10.2307/3003237. [37] M. Xue, J. Zhang, W. Tang and R. Dai, Quality improvement and pricing with reference quality effect, Journal of Systems Science and Systems Engineering, 26 (2017), 665-682.  doi: 10.1007/s11518-017-5331-y.

show all references

##### 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] R. Chenavaz, Dynamic pricing, product and process innovation, European Journal of Operational Research, 222 (2012), 553-557.  doi: 10.1016/j.ejor.2012.05.009. [3] K. Choi, R. Narasimhanb and S. W. Kim, Opening the technological innovation black box: The case of the electronics industry in Korea, European Journal of Operational Research, 250 (2016), 192-203.  doi: 10.1016/j.ejor.2015.08.054. [4] G. M. Erickson, Dynamic Models of Advertising Competition, Second Edition, Dordrecht, Kluwer, Part of the International Series in Quantitative Marketing book series, 13, 2003. doi: 10.1007/978-1-4615-1031-4. [5] G. M. Erickson, A differential game model of the marketing-operations interface, European Journal of Operational Research, 211 (2011), 394-402.  doi: 10.1016/j.ejor.2010.11.016. [6] G. Feichtinger, R. Hartl and S. Sethi, Dynamic optimal control models in advertising: Recent developments, Management Science, 40 (1994), 195-226. [7] G. E. Fruchter, Signaling quality: Dynamic price-advertising model, Journal of Optimization Theory and Applications, 143 (2009), 479-496.  doi: 10.1007/s10957-009-9575-7. [8] L. Flach and M. Irlacher, Product versus process: Innovation strategies of multiproduct firms, American Economic Journal: Microeconomics, 10 (2018), 236-277.  doi: 10.1257/mic.20150272. [9] P. De Giovanni, Quality improvement vs. advertising support: Which strategy works better for a manufacturer?, European Journal of Operational Research, 208 (2011), 119-130.  doi: 10.1016/j.ejor.2010.08.003. [10] A. Gavious and O. Lowengart, Price-quality relationship in the presence of asymmetric dynamic reference quality effects, Marketing Letters, 23 (2012), 137-161.  doi: 10.1007/s11002-011-9143-4. [11] L. Grosset and B. Viscolani, Optimal dynamic advertising with an adverse exogenous effect on brand goodwill, Automatica, 45 (2009), 863-870. [12] S. Jørgensen and G. Zaccour, Differential Games in Marketing, Dordrecht, Kluwer, Springer, Boston, MA, 2004. doi: 10.1007/978-1-4419-8929-1. [13] D. Kahnemann and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-292.  doi: 10.2307/1914185. [14] P. K. Kopalle and R. S. Winer, A dynamic model of reference price and expected quality, Marketing Letters, 7 (1996), 41-52.  doi: 10.1007/BF00557310. [15] L. Lambertini, Differential Games in Industrial Economics, Cambridge, Cambridge University Press, 2018.  doi: 10.1017/9781316691175. [16] L. Lambertini, Advertising in a dynamic spatial monopoly, European Journal of Operational Research, 166 (2005), 547-556.  doi: 10.1016/j.ejor.2004.03.013. [17] L. Lambertini and A. Mantovani, Process and product innovation by a multiproduct monopolist: A dynamic approach, International Journal of Industrial Organization, 27 (2009), 508-518.  doi: 10.1016/j.ijindorg.2008.12.005. [18] 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. [19] 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. [20] 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. [21] L. Lambertini and A. Palestini, Dynamic advertising with spillovers: Cartel vs competitive fringe, Optimal Control Applications and Methods, 30 (2009), 562-572.  doi: 10.1002/oca.881. [22] S. D. Li, Dynamic control of a multiproduct monopolist firm's product and process innovation, Journal of the Operational Research Society, 69 (2018), 714-733.  doi: 10.1057/s41274-017-0260-1. [23] P. Lin and K. Saggi, Product differentiation, process R & D, and the nature of market competition, European Economic Review, 46 (2002), 201-211.  doi: 10.1016/S0014-2921(00)00090-8. [24] G. Liu, S. P. Sethi and J. Zhang, Myopic vs far-sighted behaviours in a revenue-sharing supply chain with reference quality effects, International Journal of Production Research, 54 (2016), 1334-1357.  doi: 10.1080/00207543.2015.1068962. [25] G. W. Liu, J. X. Zhang and W. S. Tang, Strategic transfer pricing in a marketing-operations interface with quality level and advertising dependent goodwill, Omega, 56 (2015), 1-15.  doi: 10.1016/j.omega.2015.01.004. [26] A. Mantovani, Complementarity between product and process innovation in a monopoly setting, Economics of Innovation and New Technology, 15 (2006), 219-234. [27] A. Nair and R. Narasimhan, Dynamics of competing with quality-and advertising-based goodwill, European Journal of Operational Research, 175 (2006), 462-474.  doi: 10.1016/j.ejor.2005.05.015. [28] M. Nerlove and K. J. Arrow, Optimal advertising policy under dynamic conditions, Mathematical Models in Marketing, 132 (1976), 167-168.  doi: 10.1007/978-3-642-51565-1_54. [29] F. E. Ouardighi and C. S. Tapiero, Quality and the diffusion of innovations, European Journal of Operational Research, 106 (1998), 31-38.  doi: 10.1016/S0377-2217(97)00158-6. [30] F. E. Ouardighi and F. Pasin, Quality improvement and goodwill accumulation in a dynamic duopoly, European Journal of Operational Research, 175 (2006), 1021-1032.  doi: 10.1016/j.ejor.2005.06.020. [31] 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. [32] I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Operations Research, 55 (2007), 413-429. [33] S. Saha, Consumer preferences and product and process R & D, The RAND Journal of Economics, 38 (2007), 250–268. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.537.5906&rep=rep1&type=pdf. [34] G. Sorger, Reference price formation and optimal marketing strategies, Optimal control theory and economic analysis, 3 (1988), 97-120. [35] S. P. Sethi, Dynamic optimal control models in advertising: A survey, SIAM Review, 19 (1977), 685-725.  doi: 10.1137/1019106. [36] A. M. Spence, Monopoly, quality and regulation, Bell Journal of Economics, 6 (1975), 417-429.  doi: 10.2307/3003237. [37] M. Xue, J. Zhang, W. Tang and R. Dai, Quality improvement and pricing with reference quality effect, Journal of Systems Science and Systems Engineering, 26 (2017), 665-682.  doi: 10.1007/s11518-017-5331-y.
The parameters used in the numerical examples
 $\rho$ $a_0$ $a_1$ $a_2$ $a_3$ $a_4$ $\mu$ $\delta$ $\alpha$ $\beta$ $g_0$ $\theta$ $\sigma$ $q_0$ $c_0$ $\xi$ 0.06 10 0.5 0.3 0.2 0.3 0.2 0.03 7.2 5.3 20 0.05 0.02 3 15 3.9
 $\rho$ $a_0$ $a_1$ $a_2$ $a_3$ $a_4$ $\mu$ $\delta$ $\alpha$ $\beta$ $g_0$ $\theta$ $\sigma$ $q_0$ $c_0$ $\xi$ 0.06 10 0.5 0.3 0.2 0.3 0.2 0.03 7.2 5.3 20 0.05 0.02 3 15 3.9
The parameters used in the numerical examples
 $\rho$ $a_0$ $a_1$ $a_2$ $a_3$ $a_4$ $\mu$ $\delta$ $\alpha$ $\beta$ $g_0$ $\theta$ $\sigma$ $q_0$ $c_0$ $\xi$ 0.06 9 0.4 0.2 0.3 0.3 0.2 0.03 6.9 5.3 20 0.05 0.02 3 12 3.9
 $\rho$ $a_0$ $a_1$ $a_2$ $a_3$ $a_4$ $\mu$ $\delta$ $\alpha$ $\beta$ $g_0$ $\theta$ $\sigma$ $q_0$ $c_0$ $\xi$ 0.06 9 0.4 0.2 0.3 0.3 0.2 0.03 6.9 5.3 20 0.05 0.02 3 12 3.9
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