Dynamic Pricing with Stochastic Reference Price Effect

doi:10.1007/s40305-018-0227-1

Journal of the Operations Research Society of China ›› 2019, Vol. 7 ›› Issue (1): 107-125.doi: 10.1007/s40305-018-0227-1

Special Issue: Stochastic optimization

Previous Articles Next Articles

Dynamic Pricing with Stochastic Reference Price Effect

Xin Chen¹, Zhen-Yu Hu², Yu-Han Zhang³

1 Department of Industrial Enterprise and Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA;
2 NUS Business School, National University of Singapore, Singapore 119245, Singapore;
3 Two Sigma Investments, New York, NY 10036, USA

Received:2018-01-20 Revised:2018-07-21 Online:2019-03-30 Published:2019-03-30
Contact: Zhen-Yu Hu, Xin Chen, Yu-Han Zhang E-mail:bizhuz@nus.edu.sg;xinchen@illinois.edu;david8373@gmail.com
Supported by:
This research is partly supported by the National Science Foundation (Nos. CMMI-1030923, CMMI-1363261, CMMI-1538451 and CMMI-1635160), the National Natural Science Foundation of China (Nos. 71228203, 71201066 and 71520107001), and research Grant of National University of Singapore (Project R-314-000-105-133).

Abstract

Abstract: We study a dynamic pricing problem of a firm facing stochastic reference price effect. Randomness is incorporated in the formation of reference prices to capture either consumers' heterogeneity or exogenous factors that affect consumers' memory processes. We apply the stochastic optimal control theory to the problem and derive an explicit expression for the optimal pricing strategy. The explicit expression allows us to obtain the distribution of the steady-state reference price. We compare the expected steadystate reference price to the steady-state reference price in a model with deterministic reference price effect, and we find that the former one is always higher. Our numerical study shows that the two steady-state reference prices can have opposite sensitivity to the problem parameters and the relative difference between the two can be very significant.

Key words: Reference price effect, Dynamic pricing, Stochastic optimal control

Xin Chen, Zhen-Yu Hu, Yu-Han Zhang. Dynamic Pricing with Stochastic Reference Price Effect[J]. Journal of the Operations Research Society of China, 2019, 7(1): 107-125.

TrendMD

References

[1] Mazumdar, T., Raj, S.P., Sinha, I.:Reference price research:review and propositions. J. Mark. 69, 84-102(2005)
[2] Briesch, R., Krishnamurthi, L., Mazumdar, T., Raj, S.:A comparative analysis of reference price models. J. Consum. Res. 24(2), 202-214(1997)
[3] Greenleaf, E.A.:The impact of reference price effects on the profitability of price promotions. Mark. Sci. 14, 82-104(1995)
[4] Kopalle, P.K., Rao, A.G., Assunção, J.L.:Asymmetric reference price effects and dynamic pricing policies. Mark. Sci. 15(1), 60-85(1996)
[5] Popescu, I., Wu, Y.:Dynamic pricing strategies with reference effects. Oper. Res. 55(3), 834-848(2007)
[6] Hu, Z., Chen, X., Hu, P.:Dynamic pricing with gain-seeking reference price effects. Oper. Res. 64(1), 150-157(2016)
[7] Nasiry, J., Popescu, I.:Dynamic pricing with loss-averse consumers and peak-end anchoring. Oper. Res. 59(6), 1361-1368(2011)
[8] Chen, X., Hu, P., Hu, Z.:Efficient algorithms for dynamic pricing problem with reference price effect. Manag. Sci. 63(12), 4389-4408(2017)
[9] Fibich, G., Gavious, A., Lowengart, O.:Explicit solutions of optimization models and differential games with nonsmooth (asymmetric) reference-price effects. Oper. Res. 51(5), 721-734(2003)
[10] Wang, Z.:Intertemporal price discrimination via reference price effects. Oper. Res. 64(2), 290-296(2016)
[11] Krishnamurthi, L., Mazumdar, T., Raj, S.P.:Asymmetric response to price in consumer brand choice and purchase quantity decisions. J. Consum. Res. 19(3), 387-400(1992)
[12] Rajendran, K.N., Tellis, G.J.:Contextual and temporal components of reference price. J. Mark. 58, 22-34(1994)
[13] Øksendal, B.:Stochastic Differential Equations:An Introduction with Applications. Springer, New York (2002)
[14] Allenby, G.M., Rossi, P.E.:Marketing models of consumer heterogeneity. J. Econom. 89(1), 57-78(1998)
[15] Cox, J.C., Ingersoll Jr., J.E., Ross, S.A.:A theory of the term structure of interest rates. Econometrica 53, 385-407(1985)
[16] Heston, S.L.:A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327-343(1993)
[17] Bell, D.R., Lattin, J.M.:Looking for loss aversion in scanner panel data:the confounding effect of price-response heterogeneity. Mark. Sci. 19, 185-200(2000)
[18] Miranda, M.J., Fackler, P.L.:Applied Computational Economics and Finance. MIT Press, Cambridge (2004)
[19] Petruzzi, N., Dada, M.:Pricing and the newsvendor problem:a review with extensions. Oper. Res. 47, 183-194(1999)
[20] Hu, Z., Nasiry, J.:Are markets with loss-averse consumers more sensitive to losses? Manag. Sci. 64(3), 1384-1395(2018)
[21] Kloeden, P.E., Platen, E.:Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)
[22] Alfonsi, A.:On the discretization schemes for the cir (and bessel squared) processes. Monte Carlo Methods Appl. 11(4), 355-384(2005)
[23] Kounta, M., Lefebvre, M.:On a discrete version of the cir process. J. Differ. Equ. Appl. 19(8), 1276-1291(2013)
[24] Federgruen, A., Heching, A.:Combined pricing and inventory control under uncertainty. Oper. Res. 47(3), 454-475(1999)
[25] Chen,X., Simchi-Levi, D.:Coordinating inventorycontrol and pricing strategies with random demand and fixed ordering cost:The finite horizon case. Oper. Res. 52, 887-896(2004a)
[26] Chen,X., Simchi-Levi, D.:Coordinating inventorycontrol and pricing strategies with random demand and fixed ordering cost:the infinite horizon case. Math. Oper. Res. 29, 698-723(2004b)
[27] Chen, X., Simchi-Levi, D.:Coordinating inventory control and pricing strategies:the continuous review model. Oper. Res. Lett. 34(3), 323-332(2006)
[28] Chen, X., Simchi-Levi, D.:Pricing and inventory management. The Oxford handbook of pricing management, pp. 784-824(2012)
[29] Chen, X., Hu, P., Shum, S., Zhang, Y.:Dynamic stochastic inventory management with reference price effects. Oper. Res. 64(6), 1529-1536(2016)
[30] Fleming, W.H., Rishel, R.W.:Deterministic and Stochastic Optimal Control. Springer, Berlin (1982)

Dynamic Pricing with Stochastic Reference Price Effect

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 1

Recommended Articles

Metrics

Comments