Dynamic pricing for inventories with reference price effects

Chenavaz, Régis; Paraschiv, Corina Working Paper Dynamic pricing for inventories with reference price effects Economics Discussion Papers, No. 2018-50 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Chenavaz, Régis; Paraschiv, Corina (2018) : Dynamic pricing for inventories with reference price effects, Economics Discussion Papers, No. 2018-50, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/179970 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Discussion Paper No. 2018-50 | June 15, 2018 | http://www.economics-ejournal.org/economics/discussionpapers/2018-50 Dynamic pricing for inventories with reference price effects Régis Chenavaz and Corina Paraschiv Abstract This article presents a dynamic pricing model of a retailer selling an inventory, accounting for consumer behavior. The authors propose an optimal control model, maximizing the intertemporal profit with consumers sensitive to the selling price and to a reference price. The optimal dynamic pricing policy is solved with Pontryagin’s maximum principle with a structural (general) demand function. They obtain an original pricing rule, which explicitly accounts for the impact of price and inventory on future profits. The dynamics of price do not have to imitate the dynamics of the reference price. Instead, the dynamics of price are tied to opposing effects linked to this reference price. The authors also discuss managerial implications with regards to behavioral pricing policies. JEL C61 D03 D40 M21 M37 Keywords Dynamic pricing; inventory; reference price; behavioral pricing; optimal control Authors Régis Chenavaz, Kedge Business School, Aix-Marseille School of Economics, CNRS & EHESS, France, [email protected] Corina Paraschiv, Paris Descartes University & Institut Universitaire de France, Paris, France Citation Régis Chenavaz and Corina Paraschiv (2018). Dynamic pricing for inventories with reference price effects. Economics Discussion Papers, No 2018-50, Kiel Institute for the World Economy. http://www.economics-journal.org/economics/discussionpapers/2018-50 Received May 12, 2018 Accepted as Economics Discussion Paper June 8, 2018 Published June 15, 2018 © Author(s) 2018. Licensed under the Creative Commons License - Attribution 4.0 International (CC BY 4.0) 1 Introduction Dynamic pricing is a common practice where sellers adjust prices for products or services based on current demand. Developed initially by the airline industry with yield management, dynamic pricing is now a wide-spread phenomenon (Den Boer,2015). Recent technological evolutions concerning real-time access to demand-related data (big data), real-time data processing, pricing algorithm automatization (allowing for real-time computation of optimal prices), and new technologies to implement frequent price changes (smart labels) contribute to dynamic pricing diffusion in the retailing industry. A retailer is traditionally confronted with inventory management issues, for which price is a main control (Goyal and Giri,2001;Elmaghraby and Keskinocak,2003;Bakker et al., 2012). New dynamic pricing models need to be developed that also address inventory management issues together with actual consumer behavior. This point motivates our research. In this article, we study the dynamic pricing policy of a retailer selling an inventory when the reference price of consumers plays a role. In the modeling, we assume a fixed inventory without possibility of replenishment. This assumption describes well a wide range of retail products: markdown products, products at the end of their life cycle with excess inventory, or products with a short life cycle and long procurement delays (Goyal and Giri,2001;Elmaghraby and Keskinocak,2003;Bakker et al.,2012). In such situations, the retailer needs to make pricing decisions given a fixed amount of inventory and a limited time horizon. Concerning consumer behavior, we posit a reference price effect. Consumer decisions to purchase a product depend not only on the retail (or selling) price, but also on its reference price. The reference price is a psychological price used as a benchmark against which consumers compare the retail price (Sorger, 1988;Mazumdar et al.,2005). A retail price below the reference price is perceived by consumers as a gain, which increases demand. Conversely, a retail price above the reference price is perceived as a loss, reducing demand (Kopalle et al.,1996;Popescu and Wu,2007;Nasiry and Popescu,2011). The literature on behavioral dynamic pricing and inventory management inform this research. The dynamic pricing research stream that formally accounts for reference dependence was initiated decades ago by Sorger (1988) and Kopalle and Winer (1996). Building on this approach, Kopalle et al. (1996) and Fibich et al. (2003) consider asymmetric reference effects. The joint study of reference effects with inventories is more recent, with Taudes and Rudloff (2012) proposing a two-period model. Stochastic inventory is introduced in Li et al. (2015)Dye and Yang (2016) and Xue et al. (2016) study the optimal pricing of deteriorating items in the presence of a reference price. Demand depending on inventory is analyzed by Lu et al. (2016) and Hsieh and Dye (2017). Previous research uses parametric (specific) demand functions. By contrast, we adopt the approach of Popescu and Wu (2007), Chenavaz (2016), and Chenavaz (2017), using structural (general) demand functions, which enables us to derive more general results. Building on the aforementioned studies, we propose a dynamic pricing policy of a non-perishable inventory with a reference-dependent demand. To the best of our knowledge, this is the first attempt to integrate a general demand function–assuming little restriction on consumer behavior–in inventory management. Analytical results show how the price setting is tied to inventory and 2 reference price considerations. They also characterize how the dynamics of the selling price are tied to the dynamics of the reference price. A firm ignoring the psychological element would set an inadequate pricing policy, thus losing profit. The integration of consumer behavior provides a greater understanding of firm dynamic pricing strategies. 2 Modeling Assumptions We study the optimal intertemporal pricing strategy of a retailer in a monopoly situation. The retailer sells an inventory during the fixed and finite planning period T > 0. The time t∈[0, T] is continuous. 2.1 Consumer Reference Price To decide on a purchase decision at time t, a consumer compares the current retail price p(t)⩾0 with a reference price r(t)⩾0. The reference price represents a benchmark informing the customers about the opportunity of a transaction (Sorger,1988;Mazumdar et al.,2005). If p(t)> r(t), the consumer perceives a loss; he is less likely to buy. Conversely, if p(t)⩽r(t), the consumer perceives a gain; he is more likely to buy. The reference price is usually formalized through a weighted average of past product prices (Sorger,1988;Kopalle et al.,1996;Kopalle and Winer,1996; Fibich et al.,2003). We define the reference price at time t,r(t) with an exponentially decaying function: r(t) = e−βt(r0+βRt 0eβsp(s)ds) where r0is the initial reference price at time t= 0, and β⩾0 is the continuous forgetting (or adjustment speed) parameter. Differentiate r(t) with respect to time t: dr(t) dt =β(p(t)−r(t)),with r(0) = r0.(1) Equation (1) states that the variation of the reference price increases with the adjustment speed βand with the difference between the retail price p(t) and the reference price r(t). 2.2 General Reference-Dependent Demand The demand D⩾0 depends on the retail price pand the reference price r. We consider in this article a general reference-dependent demand function D=D(p(t), r(t)). This general form offers the advantage of allowing for nonlinearities when capturing dynamics related to reference price variations (Popescu and Wu,2007). For brevity and in order to simplify equations, function parameters will be further omitted, if there is no confusion. The demand function is twice continuously differentiable, satisfying the following conditions ∂D ∂p <0,∂D ∂r >0,∂2D ∂p∂r ⩽0.(2) The first condition captures a price effect (∂D/∂p < 0) by imposing that the demand decreases with the retail price. The second condition refers to a direct reference effect (∂D/∂r > 0), where the demand increases with the reference 3 price. The third condition captures an indirect reference effect (∂2D/∂p∂r ⩽0). That is, the decrease of the demand due to an increase in price is higher when the reference price is high. In line with the general demand approach initiated by Popescu and Wu (2007) and followed by Chenavaz (2016), condition (2) impose relatively weak restrictions on the impact of retail price and reference price on demand. Also such assumptions cope with the two main cases of demand separability. •Additively separable case D=f(p) + g(r), of which the classical linear demand function D=α−δp −γ(p−r) is an example. Such a function is widely used, for instance by Sorger (1988), Kopalle et al. (1996), Kopalle and Winer (1996), Dye and Yang (2016), Xue et al. (2016), and Hsieh and Dye (2017). The implication of additive separability is to impede any indirect reference effect as ∂2D/∂p∂r = 0. Such an implication simplifies the tractability of any model and thus its analysis, though at a cost of lesser generality of the results. •Multiplicatively separable case D=f(p)g(r), for which the Cobb-Douglas (isoelastic) demand function D=αp−δrγrepresents an instance. Because of opposite monotonies, the indirect reference effect writes ∂2D/∂p∂r < 0, enriching the analysis with a cross effect at the demand level. Popescu and Wu (2007) provide greater details on the properties of the demand functions. 2.3 Inventory Firm inventory at twrites I(t). We exclude the possibility of replenishment as in Taudes and Rudloff (2012), Dye and Yang (2016), Xue et al. (2016), and Hsieh and Dye (2017). The lifetime Tof the inventory is fixed, that is, there is a predeterministic selling period (say several months or seasons). Inventory decrease depends on previous sales and items do not perish over time. This situation corresponds to class (1) in the great survey by Bakker et al. (2012, p. 276). Formally, the inventory at time t, denoted I(t), is given by I(t) = I0−Zt 0 D(p(s), r(s))ds , with I(0) = I0and I(T) = 0.(3) The inventory at time tequals the initial inventory minus the cumulative demand during the period 0 to t. The differentiation of I(t) with respect to time tyields dI(t) dt =−D(p, r).(4) Equation (4) states that inventory variation at time tdecreases with current demand. Also inventory does not decrease autonomously as in the case of perishable items. 4 2.4 Current Profit The current profit of the retailer π∈Rcorresponds to the difference between revenues and costs. Revenues depend on price and demand. Costs depend on inventory level Iand unitary inventory cost h. Thus, the profit of the retailer writes π(p(t), r(t)) = p(t)D(p(t), r(t)) −hI(t).(5) Substituting (3) in (5) gives π(p(t), r(t)) = p(t)D(p(t), r(t)) −hI0−Zt 0 D(p(s), r(s))ds.(6) Equation (6) states that current profit of the retailer increases with price, current demand, and also with past demand, which, by reducing current inventory, reduces current inventory costs. 3 Dynamic Pricing Policy The retailer, in a monopoly situation, is modeled within an optimal control framework. Table 1provides the notations. Table 1: Notations T= fixed terminal time of the planning horizon, ρ= interest rate, p(t) = selling price at time t(control variable), r(t) = reference price at time t(state variable), I(t) = inventory at time t(state variable), D(p, r) = current demand, dr(t) dt =β(p−r) = reference point dynamics at time t, I(t) dt =−D(p, r) = inventory dynamics at time t, λr(t) = current-value adjoint variable for rat time t, λI(t) = current-value adjoint variable for Iat time t, h= inventory cost, π(p, r) = pD −hI = current profit, H(p, a, r, λ) = current-value Hamiltonian. 3.1 General Case The retailer maximizes the intertemporal profit by setting the optimal pricing, while accounting for reference price and inventory dynamics. With the discount 5 rate ρ⩾0, the problem of the retailer writes max p(t)⩾0ZT 0 e−ρtπ(t)dt, subject to dr(t) dt =β(p(t)−r(t)),with r(0) = r0, dI(t) dt =−D(p, r),with I(0) = I0. The maximum principle allows solving the dynamic optimization problem. With the shadow price of the reference price λr(t) and the shadow price of the inventory λI(t), the current-value Hamiltonian His H(p, r, λr, λI) = pD(p, r)−hI0−Zt 0 D(p, r)ds+λrβ(p−r)−λID(p, r).(7) The interpretation of current-value Hamiltonian His as follows: Hsums (1) current profit pD(p, r)−hI and (2) future profits due to reference point variation λrβ(p−r) and inventory variation −λID.Hthus represents the instantaneous total (intertemporal) profit at time t. The maximum principle imposes the dynamics of λIto satisfy dλI dt =ρλI−∂H ∂I =ρλI+h, with λI(T) = 0.(8) Integrating (8) with respect to time gives1 λI(t) = −ZT t e−ρ(s−t)hds. (9) By computing the integral, (9) becomes λI(t) = −(1 −e−ρ(T−t))h ρ.(10) From (9) we derive λI(t)⩽0, meaning that the shadow price of the inventory is negative over the planning period. Thus, an inventory is always costly for the retailer. Note that λI(t) is not reference dependent. Instead, it depends on the unitary inventory cost h, the interest rate ρ, and the remaining selling period T−t. Moreover, the function increases over time. The rational is the following: At the beginning of the selling period, the inventory is large, and the high inventory costs reduce the profit of the retailer. But, as time passes, the inventory is lower, and the smaller inventory costs exert less influence on profit. The maximum principle dictates the dynamics of λr dλr dt =ρλr−∂H ∂r = (ρ+β)λr−(p+h−λI)∂D ∂r ,with λr(T)=0.(11) Integrating (11) with respect to time gives2 λr(t) = ZT t e−(ρ+β)(s−t)(p+h−λI)∂D ∂r ds. (12) 1The proof of (9) is in Appendix A.1 2The proof of (9) is in Appendix A.2 6 Because λI(t)⩽0, then (p+h−λI)>0. Thus and recalling ∂D ∂r ⩾0, we conclude λr⩾0. At any time t, an increase in the reference price augments the intertemporal profit. That is, a higher reference price is associated with larger intertemporal profits of the retailer (consumers are more likely to purchase). We confine our interest to an interior solution for the price, provided it exists. The necessary and sufficient first- and second-order conditions for pimpose for all t∈(0,∞): ∂H ∂p = 0 =⇒(1 + h)D+p∂D ∂p +βλr−∂D ∂p λI= 0.(13a) ∂2H ∂p2<0 =⇒ −(2 + h)∂D ∂p −(p−λI)∂2D ∂p2>0.(13b) The first-order condition on price (13a) yields a first result.3 Proposition 1. At any time, price setting is such that −∂D ∂p p D= 1 + h+1 Dβλr−∂D ∂p λI, for all tin [0, T ]with λr⩾0and λI⩽0given by (10)and (12). Proof. Divide (13a) by Dand rearrange. Proposition 1shows how the pricing strategy of a retailer integrating inventory and reference price considerations differs from the pricing of a standard static monopoly, for which the price elasticity of demand is unitary (−(∂D/∂p)/(p/D) = 1). When the price elasticity of demand is greater than one, the demand is called elastic, and the retailer loses market power. Alternatively, when the price elasticity of demand is lower than one, the demand is called inelastic. According to Proposition 1, the price elasticity of retailer demand relies on three competing effects: one tied to current profit and two tied to the dynamics of future profits. The presence of inventory cost hhas a direct positive effect on the price elasticity of the demand. The demand becomes elastic, reflecting a loss of market power for the retailer because of the reduction of the current profit due to increased current costs. A greater reference effect also weakens the market power of the firm. The past prices set by the retailer have more influence on the current retail price. That is, the retailer has less freedom to set prices, and the demand becomes more elastic. Formally, the price elasticity of demand increases when the shadow price of the reference price λrincreases. A dynamic effect of the inventory, tied to λI, is also observed. Indeed, due to the future costs of the inventory, the retailer has an incentive to sell more in the present. This strategy enables diminishing future inventory, reducing the associated costs, and increasing future profits. Thus, the shadow price of the inventory exerts a negative impact, reducing the price elasticity of the demand. Eventually, the larger the demand D, the lower is the impact of the shadow prices of reference price and inventory on price setting. Depending on the relative strength of these competing effects, the pricing policy is such that the demand is inelastic, unitary, or elastic. 3The proof of (13a) is obvious and the proof of (13b) appears in Appendix A.3. 7 The managerial implications are straightforward. A firm considering only the current profit, because it ignores lasting reference price and inventory effects, sets a price such that the demand is elastic (1 + h). By contrast, a firm integrating current and future profits, because it accounts for lasting effects, may charge a price for which the demand is inelastic (if the lasting inventory effect outweigh the lasting reference effect and the inventory cost effect). Proposition 1, informing about the price at any time toffers preliminary insights. As such, it characterizes the optimal relationship between the price level on the one side and inventory cost and dynamics effects on the other side. Yet, it says nothing about the conditions under which the optimal price goes up or down after a change in the inventory cost and dynamics elements. That is, Proposition 1maintains silence about any causal relationship. Proposition 1has to hold during the whole selling period. To maximize the intertemporal profit, marginal revenue variations must balance marginal cost variations. The relationship between the dynamics of price and the intertemporal elements is made explicit by the time differentiation of the first-order conditions (13a). This time decomposition method has been used to show explicit causality (see, for example, Kalish (1983) and Chenavaz and Jasimuddin (2017)). The first-order condition on price (13a) must hold at any time t∈[0, T ]. Its decomposition with respect to time supports the following result. Proposition 2. Over time, price dynamics are such that dp dt     −(2 + h)∂D ∂p −(p−λI)∂2D ∂p2 | {z } +    =dr dt    (1 + h)∂D ∂r | {z } + + (p−λI)∂2D ∂p∂r | {z } −     +β   (ρ+β)λr | {z } + −(p+h−λI)∂D ∂r | {z } −     −∂D ∂p  ρλI+h | {z } +  , for all tin (0, T )with λr⩾0and λI⩽0given by (10)and (12). Proof. Differentiate (13a) with respect to time t. The detailed proof is in Appendix A.4. Proposition 2captures the evolution of the optimal price over time. The second order condition (13b) guarantees that, on the left side of equation (2), the second factor is positive. On the right-side of equation (2) appear three additively separable terms. The first term relates to reference price dynamics, the second to the adjustment speed of the reference price, and the third to the price sensitivity of demand. For the first two terms the impact is ambiguous as they result from two competing effects. The third term has a positive impact,4 4The proof is straightforward. Substituting λIfrom (10) yields ρλI+h=−h(1 − e−ρ(T−t)) + h=he−ρ(T−t), which is positive. That is, ρλI+h > 0. Further, demand conditions (2) impose ∂D ∂p <0, enabling derivation of the positivity of the third factor: − ∂D ∂p (ρλI+h)>0. 8 Please note: You are most sincerely encouraged to participate in the open assessment of this discussion paper. You can do so by either recommending the paper or by posting your comments. Please go to: http://www.economics-ejournal.org/economics/discussionpapers/2018-50 The Editor © Author(s) 2018. Licensed under the Creative Commons License - Attribution 4.0 International (CC BY 4.0).