Price Trends in a Dynamic Pricing Model with Heterogeneous Customers: A Martingale Perspective

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1 OPERATIONS RESEARCH Vol. 57, No. 5, September October 2009, pp in X ein inform doi /opre INFORMS TECHNICAL NOTE INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(). Additional information, including right and permiion policie, i available at Price Trend in a Dynamic Pricing Model with Heterogeneou Cutomer: A Martingale Perpective Xiaowei Xu Department of Supply Chain Management and Marketing Science, Rutger, The State Univerity of New Jerey, Newark, New Jerey 07102, xiaoweix@andromeda.rutger.edu Wallace J. Hopp Stephen M. Ro School of Buine, Univerity of Michigan, Ann Arbor, Michigan 48109, whopp@umich.edu Thi note decribe probabilitic propertie of optimal price ample path in a dynamic pricing model with a finite horizon and limited tock. We aume that cutomer arrival follow a nonhomogeneou Poion proce. We how that if cutomer willingne-to-pay increae rapidly over time, then the optimal price proce follow a ubmartingale, which implie an upward price trend. Alternatively, if cutomer willingne-to-pay decreae rapidly over time, then the optimal price proce follow a upermartingale, which implie a downward price trend. Subject claification: probability: tochatic model application. Area of review: Revenue Management. Hitory: Received June 2006; reviion received January 2007, Augut 2007, February 2008, April 2008, September 2008, December 2008; accepted February Publihed online in Article in Advance Augut 12, Introduction The aim of thi note i to characterize dynamic price procee in market with an end date or a eaonal tructure. A growing literature ha addreed optimal dynamic pricing policie in uch environment. For example, Gallego and van Ryzin (1994) developed a widely ued framework, in which cutomer arrival are modeled a a Poion proce. They tudied the cae with homogeneou demand (i.e., cutomer willingne-to-pay i contant over time) and found that the optimal price i decreaing in the remaining inventory level at any given time (the inventorymonotonicity property) and alo decreaing over time with a fixed inventory level (the time-monotonicity property). Zhao and Zheng (2000) allowed nonhomogeneou demand (e.g., cutomer willingne-to-pay changing over time). They proved that the inventory-monotonicity property continue to hold and found a ufficient condition for the timemonotonicity property. The ufficient condition require a decreae in cutomer willingne-to-pay over time. However, when an optimal pricing policy i implemented, both inventory level and time change imultaneouly. A illutrated in Gallego and van Ryzin (1994) and Bitran and Mondchein (1997), an optimal price ample path uually exhibit a zig-zag hape a inventory level decreae over time. Becaue of thi, there are no monotonic reult for the optimal price proce at the ample-path level. In thi note, we tudy the optimal price proce from a new perpective. Specifically, we offer condition under which a monotonic price trend i generated by an optimal pricing policy in the type of environment conidered by Gallego and van Ryzin (1994) and Zhao and Zheng (2000). However, the trend to which we refer i decribed in a probabilitic ene. That i, intead of focuing on the ample-path level, we conider the monotonicity of the optimal price proce in expectation. We how that if cutomer willingne-to-pay increae rapidly over time, then the optimal price proce follow a ubmartingale, which implie an upward price trend. Alternatively, if cutomer willingne-to-pay decreae rapidly over time, then the optimal price proce follow a upermartingale, which implie a downward price trend. 2. The Model We aume that a firm ha a tock level of n (a nonnegative integer) item at time 0, which it ell over the interval 0 T. Cutomer arrival follow a nonhomogeneou Poion proce with intenity, where 0 T. We aume that cutomer, who arrive at time, havea utility function V p = U p and purchae the product if their utility value V p i larger than zero, where U i the reervation value of the product, i the price enitivity, and p i the product price. Notice that cutomer willingne-to-pay can be dynamic over time a a reult of two effect: (a) cutomer price enitivity 1298

2 Xu and Hopp: Price Trend in a Dynamic Pricing Model Operation Reearch 57(5), pp , 2009 INFORMS 1299 INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(). Additional information, including right and permiion policie, i available at change over time, and (b) the reervation value U change over time. We aume that cutomer are heterogeneou in their reervation value of the product; that i, U i a random variable with probability ditribution u. We let u be the denity function of u and define the failure rate of u a h u = u / u, where u = 1 u. Thi implie that the total demand rate at time given price p i p = p. We aume that p i twice differentiable in p and let = p 0 T be a nonanticipatory pricing policy that control the actual cutomer demand proce M. Notice that if = 1 for 0 T, then we have the demand proce tudied in Zhao and Zheng (2000). Condition 1. For every 0 T, p i log-concave in p, or equivalently, h u i increaing in u 0 +. Condition 1 mean that U ha an increaing failure rate for every 0 T (ee Lariviere 2006 and Ziya et al for application of failure rate in upply chain management and revenue management). By Condition 1, p = log p i concave in p. At time, with remaining inventory level l, the firm maximize expected revenue J l = E T p t dm t ubject to the inventory contraint T dm t l, where J l T = 0 and J 0 = 0 for l n and 0 T. Notice that the inventory contraint can be atified by etting price to infinity after a tockout (called the null price in Gallego and van Ryzin 1994). We denote an optimal pricing policy by arg max J l, where repreent the et of all nonanticipatory pricing policie atifying the inventory contraint. Thi allow u to expre the optimal expected revenue in T given inventory level l a J l = J l = up J l. By the Principle of Optimality, we obtain the Hamilton- Jacobi-Bellman equation for J, J where J l = up p p I l p l = J l I l = J l J l 1 1 l n and 0 T (ee derivation detail in Gallego and van Ryzin 1994 and Zhao and Zheng 2000). By Condition 1, there exit a unique optimal olution p l for the right ide of the above equation. Hence, the optimal pricing policy i given by = p l 1 l n 0 T. Although and J mut be computed numerically for mot cae, there exit an implicit tructure between and J, which enable u to prove a martingale property for the optimal price proce by applying Dynkin formula (Rolki et al. 1999). Dynkin formula ha been ued by other to obtain optimal expected revenue function (ee, e.g., Feng and Xiao 2000, Feng and Gallego 2000). Let N be a nonhomogeneou Markovian proce, which model the inventory proce driven by the optimal pricing policy (ee detail in Online Appendix I, which i available a part of the online verion that can be found at and F N be the -algebra generated by N t, t. We define an optimal topping time = inf 0 T N = 0, which repreent the time when the inventory level fall to zero, and the optimal price proce P = { p N if < p 1 if Becaue the optimal pricing policy can take an arbitrarily defined value when a tockout occur, we top the price proce P immediately after tockout occur and let the price proce take the value it had jut before a tockout occur. In Online Appendix II, we tudy an alternative price proce, in which we top the price proce when inventory level fall to one, and how that imilar monotonicity reult to thoe preented below alo hold for thi alternative price proce under weaker condition Upward Price Trend We firt etablih that the optimal price proce i a ubmartingale, and hence the price trend i upward under the two condition tated below. Condition 2. For every 0 T, 1/h u i convex in u. Condition 2 hold for uniform, logitic, and normal ditribution and gamma and Weibull ditribution with hape parameter no le than one. Thee ditribution alo have increaing failure rate, and hence atify Condition 1 (proof of thi and all other reult are available in Online Appendix I). We offer an economic interpretation of Condition 2 by adopting an idea ued in Tyagi (1999). We define the marginal revenue function mr p = p p p = p + p / p = p 1 h p Thi implie that 2 mr p p 2 [ = 2 1 u 2 h u ] u= p Hence, Condition 2 i equivalent to the condition that the marginal revenue function mr p i concave in p.

3 Xu and Hopp: Price Trend in a Dynamic Pricing Model 1300 Operation Reearch 57(5), pp , 2009 INFORMS INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(). Additional information, including right and permiion policie, i available at Let à = up u u T u and B = up 0 T. By the Markov inequality, à E U T. Let G = up u u T u 0 = arg max u 0 u u u u We define H 1 = B G and H 2 = BÃ. where Condition 3. Either (1) up 0 T log < H 1 and h u i decreaing in for every u; or (2) i decreaing in, h u i decreaing in for every u, and inf u u0 + 0 T / 1/h u > H 2. If h u = h u (i.e., cutomer reervation value of the product i independent of time), then Condition 3.1 implie that price enitivity i rapidly decreaing in. If i contant over time, Condition 3.2 implie that U 1 hr U 2, where T and hr i the hazard rate order. By Theorem of Müller and Stoyan (2002), U 1 t U 2, where t i the uual tochatic order. Hence, Condition 3.2 implie that cutomer reervation value i rapidly increaing over time. Notice that 2 log p p = h p = h u p p + h p h p > 0 where the lat inequality follow from Condition 1 and 3. Hence, p i log-upermodular in p. Thi implie that p H / p L = p H / p L i increaing in, where p H >p L. Zhao and Zheng (2000) made a imilar log-ubmodularity aumption (Aumption 1 therein) on p to etablih the time-monotonicity property of the optimal pricing policy. By their interpretation, p H / p L i the conditional probability that a cutomer i willing to pay a higher price (p H ) given that he would like to buy at a lower price (p L ). With thi interpretation, Condition 3 implie that cutomer willingne-to-pay i rapidly increaing over time becaue Condition 3 require that p i ufficiently log-upermodular in p. Finally, we notice that a rapid increaing of willingne-to-pay i a conequence of two effect: (a) cutomer price enitivity i rapidly decreaing over time, and (b) reervation value of the product i rapidly increaing over time. Theorem 1. If Condition 1 3 hold, then P i a F N -ubmartingale. In particular, E P i increaing in 0 T. Hence, Condition 1 3 repreent ufficient condition for the optimal price path to be increaing in expectation. Example 1. We aume that u i independent of time (i.e., u = u ), u ha an increaing failure rate h u, and 1/h u i convex in u. Hence, Condition 1 and 2 are atified. Let = e K and p = p. IfK< H 1, then Condition 3.1 hold. Becaue u = u, à = up u u u, and G = up u u0 + u u, where u 0 = arg max u u u. If u i an exponential ditribution with mean one, then à = e 1, G = e 1, and u 0 = 1. Hence, H 1 = e 1 B. Example 2. Let = 1 for 0 T. We aume that u i a uniform ditribution on U U for every, where U>0and U i increaing in. Becaue 1/h u = U u, if inf 0 T U > H 2, then Condition 3.2 i atified. To calculate H 2, we notice that à = U T 2 / 4 U T U. A pointed out by Bitran and Mondchein (1997) and Zhao and Zheng (2000), traveler are willing to pay more for their fare a the departure time approache. By Theorem 1, we would expect an optimal pricing policy to exhibit an upward trend during the eaon if cutomer willingne-to-pay increae rapidly over time Downward Price Trend In thi ection, we identify condition that guarantee a downward price trend. We how that the condition require a rapid drop of cutomer willingne-to-pay over time. Let A = up u u 0 u. By the Markov inequality, A E U 0. Let [ ] 1 F = inf and 0 T u h u T G = up u u u T + 0 T where u T = arg max u u T u. Notice that F 0 under Condition 1. We define H 1 = AB 2 G 1 F T and H 2 = AB 2 1 F T. Condition 4. Either (1) inf 0 T log >H 1 and h u i increaing in for every u, or (2) i increaing in, h u i increaing in for every u and up u ut + 0 T / 1/h u < H 2. Becaue 2 log p p = [ h u p p + h p ] h p Condition 4 implie that p i ufficiently logubmodular in p ; that i, cutomer willingne-to-pay decline rapidly during the eaon. Condition 4.1 repreent a harp increae in price enitivity and Condition 4.2 a ubtantial decreae in cutomer reervation value over time.

4 E[P()] Xu and Hopp: Price Trend in a Dynamic Pricing Model Operation Reearch 57(5), pp , 2009 INFORMS 1301 INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(). Additional information, including right and permiion policie, i available at Theorem 2. If Condition 1, 2, and 4 hold, then P i a F N -upermartingale. In particular, E P i decreaing in 0 T. Hence, Condition 1, 2, and 4 repreent ufficient condition for the optimal price path to be decreaing in expectation. Example 3. We aume that u i independent of time (i.e., u = u ), u ha an increaing failure rate h u and 1/h u i convex in u. Let = e K and p = p. IfK>H 1, then Condition 4.1 hold. Becaue u = u and G = up u u T + u A = up u u u F = u 1/h u T where u T = arg max u u u. If u i an exponential ditribution with mean one, then A = e 1, F = 0, G = e 1, and u T = 1. Hence, H 1 = e 2 B 2 T. Figure 1. (a) E[P()] The mean of the optimal price proce P. 0 (c) E[P()] (b) Example 4. Let = 1 for 0 T. We aume that u i a uniform ditribution on U U for every, where U>0 and U i decreaing in. Becaue 1/h u = U u, ifup 0 T U < H 2, then Condition 4.2 i atified. To calculate H 2, we notice that A = U 0 2 / 4 U 0 U and F = 1. A pointed out by Bitran and Mondchein (1997) and Zhao and Zheng (2000), mot early adopter are willing to pay more for fahion good. Theorem 2 implie that if thi decreae in willingne-to-pay i ufficiently large, the optimal price ample path will be decreaing in an expectation ene. 3. A Numerical Example We reviit the numerical example in the bottom of Figure 1 in Gallego and van Ryzin (1994, p. 1005). Let T = 1, n = 25, and p = 100e p, where = e K and K We dicretize time interval 0 T into 0 2 L, where L = T, and imulate the actual cutomer demand on j j + 1 by a Bernoulli random variable with mean p j j, where j = 0 L 1 (ee Note. (a) K = 4; (b) K = 0; (c) K = 4. 0

5 Xu and Hopp: Price Trend in a Dynamic Pricing Model 1302 Operation Reearch 57(5), pp , 2009 INFORMS INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(). Additional information, including right and permiion policie, i available at detail of Markov chain imulation in Kuhner and Dupui 2001). Figure 1 how the mean of the optimal price proce P. Figure 1(a) how an increae in optimal expected price when the price enitivity function decreae rapidly, which i conitent with Theorem 1. In contrat, Figure 1(c) how a decreae in optimal expected price when the price enitivity function increae rapidly, which i conitent with Theorem 2. Notice that Condition 3 and 4 are ufficient condition for a monotonic trend of the optimal price proce in expectation, but are not neceary condition (e.g., K =4 doe not atify Condition 3 and 4 demontrated in Example 1 and 3). Finally, Figure 1(b) how an initial increae in optimal expected price, but a harp decreae at the end of the period. Hence, the price trend i not monotonic when the price enitivity function change lowly over time. 4. Concluion The probabilitic characterization of optimal price ample path given in thi note offer an explanation of how price trend are affected by cutomer willingne-to-pay, which in turn depend on cutomer price enitivity and reervation value of the product. A ufficiently rapid rie (decline) in willingne-to-pay implie an upward (downward) price trend. 5. Electronic Companion An electronic companion to thi paper i available a part of the online verion that can be found at inform.org/. Online Appendix I include technical detail and all proof. Online Appendix II preent the tudy of an alternative price proce. Acknowledgment The author thank the aociate editor and two anonymou referee for their helpful comment. Reference Bitran, G., S. V. Mondchein Periodic pricing of eaonal product in retailing. Management Sci. 43(1) Feng, Y., G. Gallego Perihable aet revenue management with Markovian time dependent demand intenitie. Management Sci. 46(7) Feng, Y., B. Xiao Optimal policie of yield management with multiple predetermined price. Oper. Re. 48(2) Gallego, G., G. van Ryzin Optimal dynamic pricing of inventorie with tochatic demand over finite horizon. Management Sci. 40(8) Kuhner, H. J., P. Dupui Numerical Method for Stochatic Control Problem in Continuou Time. Springer-Verlag, New York. Lariviere, M. A A note on probability ditribution with increaing generalized failure rate. Oper. Re. 54(3) Müller, A., D. Stoyan Comparion Method for Stochatic Model and Rik. John Wiley & Son, Wet Suex, UK. Rolki, T., H. Schmidli, V. Schmidt, J. Teugel Stochatic Procee for Inurance and Finance. John Wiley & Son, New York. Tyagi, R. K A characterization of retailer repone to manufacturer trade deal. J. Marketing Re Zhao, W., Y. S. Zheng Optimal dynamic pricing for perihable aet with nonhomogeneou demand. Management Sci. 46(3) Ziya, S., H. Ayhan, R. D. Foley Relationhip among three aumption in revenue management. Oper. Re. 52(5)