A Mathematical Analysis on Daily Inventory Clearance Pricing with Consumer’s Reference Price
 Author: Koide Takeshi, Sandoh Hiroaki
 Organization: Koide Takeshi; Sandoh Hiroaki
 Publish: Industrial Engineering and Management Systems Volume 11, Issue1, p30~38, 01 March 2012

ABSTRACT
This paper discusses a clearance pricing on daily perishable products considering a reference price of consumers. The daily perishable products are sometimes sold at a discount price before closing time to stimulate demand when the number of unsold products is more than initially envisioned. The discount pricing results both in an increase of the revenue of the day and in a decrease of the disposal cost. The discounting, however, also declines a reference price of consumers which is a mental price and serves as an anchor price to judge if a current sales price is loss or gain for the consumers. An excess discounting decreases the demand for the products sold at a regular price in the future and diminishes longterm profit.
This study conducts a mathematical analysis on the clearance pricing problem for a single period with stochastic variations both on demand and on the inventory level at clearance time. The expected profit function especially depends on the response of consumers to the clearing price against their reference prices. A procedure is proposed to derive an optimal clearance price when consumers are lossneutral. A sufficient condition is shown to obtain an optimal price for lossaverse and lossseeking consumers by an analogous procedure.

KEYWORD
Optimal Pricing , Clearance , Inventory , Reference Price

1. INTRODUCTION
Many retail stores nowadays sell a range of prepared daily food items such as fried meals, cooked food, salad, and sushi. If such daily items are unsold at closing time, they are disposed or reused as ingredients for other items. It is a typical newsvendor problem to find an optimal amount of their initial supply to maximize an expected profit of the day.
Under practical circumstances, it is difficult to estimate demand function for target items accurately. When forecasted demand is more than actual one, retail stores often mark down a sales price to stimulate demand and to reduce the number of unsold items. If the discount pricing is conducted appropriately, it improves the profit of the day. The retail stores decrease disposal cost and might increase the revenue.
The retail stores, however, have to pay attention to the influence of the discount sales on consumers. Consumers’ reference price will be declined when they purchase the products at a discount price. The reference price is a mental price and plays a role as a criterion to recognize whether an asking price is gain or loss. The reduced reference price decreases forthcoming demand sold at a regular price. As a result, a discount sale one day reduces profit in the future. From a viewpoint of profitability, retail stores should determine the clearing discount price carefully with considering its influence on longterm profit.
The reference price is initially proposed as a reference point in the prospect theory by Kahneman and Tversky (1979). The reference price itself is actively researched especially in the area of marketing science. Kalyanaram and Winner (1995) summarized past research with respect to the reference price and mentioned that the reference price is generated by a series of past sales prices. Consumers react differently according to whether the sales price is less or greater than the reference prices of the consumers. The asymmetric reaction is a key concept on behavioral economics wellresearched recently.
Our study aims to derive an optimal clearance pricing on daily perishable products to maximize a longterm expected profit. This paper focuses on a single period model as a first step for the longterm optimization. A model is proposed where stochastic demand, stochastic inventory level, which means the amount of unsold items, and consumers’ reference price effect are considered. Greenleaf (1995) proposed a model for the first time to derive an optimal pricing considering the reference price effect and Kopalle
et al . (1996) extended his model. Their models, however, are for promotion planning and do not contain the concept of inventory level. Popescu and Wu (2007) showed optimal pricing policies for the promotion problem with more general type of demand function, but inventory level is also excluded from their models. The above studies treat deterministic demand functions and derive an optimal price through the dynamic programming.The problem to determine an optimal supply quantity toward uncertain demand is wellknown as newsvendor problem. The newsvendor problem is originally studied by Arrow
et al . (1951) and various models have been proposed since then. Petruzzi and Dada (1999) discussed the relationship between pricing and inventory control. They summarized past research with respect to pricing and the newsvendor problem. They introduced a model which treats stochastic demand and explores both an optimal price and an optimal inventory level, but the reference price effect is not discussed in the model.In this study, an expected profit function is formulated and analyzed mathematically. First, a profit function with deterministic demand and inventory level in a single period model is discussed. The result reveals that the shape of the profit function depends on the consumer' s attitude toward gain and loss, in other words, on whether consumers are lossneutral (LN), lossaverse (LA), or lossseeking (LS). Then, the discussed model is extended to treat stochastic demand and inventory level and a sufficient condition is shown to obtain a unique optimal price to maximize an expected profit.
2. PROBLEM DESCRIPTION
Consider a firm deals in a type of product under monopoly. The target time horizon is limited to a single period in this study. The firm prepares a certain amount of the products before opening time at a unit procurement cost
c (> 0) and starts to sell the product at a regular pricep_{H} . No replenishment is considered in this model. After closing time, unsold products are salvaged or disposed at a unit costh , which means that the unsold products are salvaged ifh < 0 and they are disposed otherwise. Lets (> 0) be a unit penalty cost for an opportunity loss.At a prescheduled time during the operating hours, the firm can discount the products to stimulate demand. This study focuses on the optimal discount pricing. The only decision variable in this model is the discount price
p in the range [p_{L} ,p_{H} ]. The firm determines the pricep before the prescheduled time in advance with considering uncertain inventory levelQ and supposed consumer’s reference pricer for the products. If some products are unsold at the prescheduled time, the unsold products are sold at pricep from then to closing time. The reference pricer exists in the range [p_{L} ,p_{H} ]. The inventory levelQ is assumed to be a random variable and it is given byQ =q +ε_{q} , whereq is the average ofQ andε_{q} is a random factor whose mean is 0 and range is [q_{L} ,q_{H} ].The demand function for the product
D (p ,r ) includes the reference price effect and stochastic variation:The positive parameters
β _{2G} andβ _{2L} respectively represent the degree of the reference price effect when consumers recognize the discount pricep as a gain (p <r ) and a loss (p >r ). The parameters characterize the consumer’s response toward the selling price. The consumers withβ _{2G} <β _{2L},β _{2G} =β _{2L}, andβ _{2G}>β _{2L} are called LA, LN, and LS, respectively. All of the parametersB _{0G}(r ),B _{0L}(r ),B _{1G},B _{1L} are consequently positive. The random factorε_{d} in the demand function, whose mean is 0 and range is [d_{L} ,d_{H} ], is assumed to be independent of both the sales pricep and reference pricer . Assume ?h <c <p_{L} andD (p ,r ) > 0 for anyp ,r ∈ [P_{L} ,P_{H} ].3. OPTIMAL PRICING IN A DETERMINISTIC CONDITION
This section discusses the optimal pricing in case that the demand and inventory level are deterministic as a simple case. In other words, we here treat the case where
ε_{d} andε_{q} are constantly equal to 0.3.1 Optimal Pricing for LossNeutral Consumers
This subsection confines our discussion to the optimal pricing for LN consumers. Let
β _{2G} =β _{2L} =β _{2} then the demand functionD (p ,r ) =d (p ,r ) is given by the following equation:The both parameters
B _{0}(r ) andB _{1} are positive. Letbe the price
p which satisfiesd (p ,r ) =q . Then, it holdsWhen the products are sold at price
p for lossneutral consumers with reference pricer and the inventory levelQ is a constantq , the profitπ (p ,q ,r ) is represented byLet
be the price
p which maximizesπ _{Λ}(p ,q ,r ). A simple analysis ofπ _{Λ}(p ,q ,r ) derivesLet p^{* }(
q ,r ) be the price to maximizeπ (p ,q ,r ), then the following theorem is proved.Theorem 1: When both the demand function D(p, r) and the inventory level Q are deterministic and consumers are LN, the optimal price p^{* }(q, r) which maximizes the profit π (p, q, r) is derived by the following equation: Proof: The functionπ _{Θ}(p ,q ,r ) monotonically increases sinceMeanwhile, the function
π _{Λ}(p ,q ,r ) is a concave function shown as follows:Hence, the profit
π (p ,q ,r ) increases monotonically in the range ofwhere
π (p ,q ,r ) =π _{Θ}(p ,q ,r ), and it is concave in the range ofwhere
π (p ,q ,r ) =π _{Λ}(p ,q ,r ). Figure 1 depicts the shape ofπ (p ,q ,r ). When the two pricesand
exits in [
p_{L} ,p_{H} ], the profit functionπ (p ,q ,r ) monotonically increases forand monotonically decreases for
Then, the greater between
and
maximizes the profit
π (p ,q ,r ). □Corollary 1: The optimal price p^{*} (q, r) by Equation (11) is expressed as follows: Proof: The above inequalities directly derive the corollary. □
Differentiating
and
with respect to
r givesHence, both
and
increase monotonically with respect to
r and the increment ofis twice that of
Figure 2 illustrates the region regarding the optimal price with assuming
and
An numerical example for the optimal price
p ^{* }(q ,r ) are shown as threedimensional images in Figure 3 under the following parameters setting:β _{0} = 100,β _{1} = 0.1,β _{2} = 0.01, andh = 50.Here, let
be the
β _{2} which satisfiesthen Equation (10) yields
Lemma 1 explains the influence of
β _{2} onπ _{Λ}(p ,q ,r ) andLemma 1: The price is decreasing with respect to β_{2}. The profit π_{Λ}(p, q, r) is increasing, constant, and decreasing with respect to β_{2} for p < r, p = r, and p > r, respectively. Furthermore, the maximum of π_{Λ}(p, q, r) is decreasing and increasing with respect to β_{2} for and respectively. Proof: Differentiating Equation (10) deriveshence
is decreasing with respect to
β _{2}. Similarly, differentiating Equation (9) derivesFrom the assumption that
p +h > 0, Equation (22) proves the property onπ _{Λ}(p ,q ,r ). Finally, differentiatingwith respect to
β _{2} givesEquations (21), (22), and (23) prove the last property regarding the maximum of profit
The profit functions
π _{Λ}(p ,q ,r ) with several values ofβ _{2} are depicted in Figure 4, wherer = 450 andSince
decreases monotonically with respect to
β _{2}, the maximum vertex ofπ _{Λ}(p ,q ,r ) moves leftward and downward and approaches toward the point (r ,π _{Λ} (r ,q ,r )) with increasingβ _{2} toThe vertex of
π _{Λ}(p ,q ,r ) moves leftward and upward from (r ,π _{Λ} (r ,q ,r )) whenβ _{2} increases from3.2 Optimal Pricing for Asymmetry Consumers
This subsection discusses the optimal pricing for LA and LS consumers, namely when it holds that
β _{2G} ≠β _{2L}. In this case, the expected demand functiond (p ,r ) can be expressed as follows:since both
d_{G} (p ,r ) andd_{L} (p ,r ) linear functions with respect top and they have a common point atp =r .In the same manner in the previous subsection, as
was defined as follows:
Figure 5 indicates the two prices
and
Since Equation (7) also holds for LA and LS consumers,
π (p ,q ,r ) =π _{Θ}(p ,q ,r ) forFrom Equation (8), the profit function
π _{Θ}(p ,q ,r ) in the case of inventory shortage is decreasing asd (p ,r ) increases, and then it is expressed aswhere
π _{ΘG}(p ,q ,r ) andπ _{ΘL}(p ,q ,r ) are theπ _{Θ}(p ,q ,r ) in Equation (8) withβ _{2} =β _{2G} andβ _{2} =β _{2L}, respectively. The functionπ _{Θ}(p ,q ,r ) in Equation (26) monotonically increases with respect top since bothπ _{ΘG}(p ,q ,r ) andπ _{ΘL}(p ,q ,r ) increase monotonically as proved by Equation (12). The profitπ (p ,q ,r ) for LA and LS consumers increases monotonically in the rangeThe profit function
π _{Λ}(p ,q ,r ) in the case of excessive inventory is expressed assince Equation (9) shows
π _{Λ}(p ,q ,r ) increases asd (p ,r ) increases. The profit functionsπ _{ΛG}(p ,q ,r ) andπ _{ΛL}(p ,q ,r ) are theπ _{Λ}(p ,q ,r ) in Equation (9) withβ _{2} =β _{2G} andβ _{2} =β _{2L}, respectively. The two profit functionsπ _{ΛG}(p ,q ,r ) andπ _{ΛL}(p ,q ,r ) have a common point onp =r . Letand
be respectively the prices on which
π _{ΛG}(p ,q ,r ) andπ _{ΛL}(p ,q ,r ) have a maximum, namelyLemma 1 concludes that it holds
LA consumers and
for LS consumers. Then, Lemma 1 restricts the possibility of the shapes of the profit functions
π _{Λ}(p ,q ,r ) for LA and LS consumers, represented in Figure 6 and Figure 7, respectively. The function for LA consumers is concave in any cases in Figure 6. The function for LS consumers is also concave except in the case ofwhen the function is bimodal. This discussion introduces the following theorem as a procedure to derive the optimal price for the asymmetry consumers.
Theorem 2: When both the demand function D(p, r) and the inventory level Q are deterministic and consumers are LA or LS, the optimal price p^{* }(q, r) which maximizes the profit π(p, q, r) is derived by the following equations: Proof: The setP _{1}^{*} consists of the candidate prices to maximizeπ _{Λ}(p ,q ,r ) without considering the lower and upper limits of the sales price. Forin the middle case in Figure 7,
π _{Λ}(p ,q ,r ) is bimodal and bothand
could be optimal. In the case of max
is the optimal price both for LA and LS consumers. Similarly,
is the optimal
when it holds min
In the last case, the middle case in Figure 6,
r is the optimal. In the same manner in Theorem 1, Equation (32) ascertains that the candidate prices inP _{1}^{*} to be optimal forπ (p ,q ,r ). □Note that the cardinality of
P _{2}^{*} is two in the case ofIn the other cases, the cardinality is one and Equation (32) explores the optimal price without Equation (30).
4. OPTIMAL PRICING IN STOCHASTIC CONDITION
This section discusses the optimal pricing in case that the demand
D (p ,r ) and inventory levelQ are stochastic.4.1 Optimal Pricing for LN Consumers
This subsection confines our discussion to the optimal pricing for LN consumers. The demand function
D (p ,r ) is defined by Equations (1) and (5). Define new variablesz =q d (p ,r ) andε =ε_{q} ε_{d} . in accordance with Petruzzi and Dada (1999). Note thatThe average of
ε is 0 and the range ofε is [q_{L} d_{H} ,q_{H} d_{L} ]. Then, the profitπ (p ,q ,r ) is expressedLet
f ( ？ ) andF ( ? ) be the probabilistic density function and the distribution function of the variable . Defineε The expected profit Π(
p ,q ,r ), hence, is obtained byThe expected profit Π(
p ,q ,r ) can be rewritten byIn Equation (36), Ψ(
p ,r ) andL (p ,z ) respectively imply the profit forQ =D (p ,r ) and the expected cost incurred by excess or deficiency of inventory. The expected volumes of excess and deficiency of inventory are denoted by Λ(z ) and Θ(z ) defined in Equations (39) and (40), respectively. The following theorem derives the optimal pricep ^{*}(q ,r ) to maximize the expected profit π(p ,q ,r ).Theorem 3: When both the demand function D(p, r) and the inventory level Q are stochastic and consumers are LN, the optimal price p^{* }(q, r) which maximizes the expected profit π( p ,q ,r ) is derived by the following equation:where
is the unique solution of g (p ,q ,r ) = 0:Proof: Differentiating from Equations (36) to (40) with respect to pyieldsUsing the assumption of
h <p_{L} , Equations (48) and (49) prove that π(p ,q ,r ) is concave with respect top and has a unique maximumwhich satisfies
g (p ,q ,r ) = 0. □Corollary 2: The optimal price p^{* }( q ,r ) by Equation (41) is expressed as follows:Proof: It is obviously proved from the concavity of the function π(p ,q ,r ). □Similarly to the deterministic case, let
be the
β _{2} which satisfiesthen
is obtained from the equation
g (r ,q ,r ) = 0:provided that
is not equal to 0. The next lemma reveals a property on Π (
p ,q ,r ) with respect toβ _{2}.Lemma 2: The expected profit Π( p ,q ,r ) is concave with respect toβ _{2}. Furthermore, under the circumstance which satisfies the following inequalityΠ(p, q, r) has the following properties: (i) Π (p, q, r) is increasing, constant, and decreasing with respect to β _{2} for p < r, p = r, and p > r, respectively; (ii) The maximum of Π (p, q, r) is decreasing and increasing with respect toβ _{2} forrespectively; (iii) is decreasing with respect to β_{2}. Proof: Differentiating from Equations (36) to (40) with respect toβ _{2} yieldsEquation (57) proves the concavity of Π(
p ,q ,r ). Equation (56) proves property (i) from the assumption thatp +h > 0. The properties (ii) and (iii) are proved from property (i) and the fact that π(r ,q ,r ) is constant for any value ofβ _{2}. □Lemma 2 indicates that the contour of the expected profit function Π(
p ,q ,r ) shown in Figure 8 which resembles that ofπ _{Λ}(p ,q ,r ) shown in Figure 4. Figure 9 depicts the fluctuation of π(p ,q ,r ) with respect toβ _{2} forp = 410, 430, 450, 470, and 490 wherer = 450. The parameters in Figure 9 are set to the same values in Figure 8. In Figure 9, π(p ,q ,r ) fluctuates as mentioned in property (i) of Lemma 2 except in the casep = 410. The functionr π(p ,q ,r ) withp = 410 decreases forβ _{2} > 0.2, where Inequality (52) does not hold.4.2 Optimal Pricing for Asymmetry Consumers
In the case of
β _{2G} ≠β _{2L} , if Inequality (52) holds, Lemma 2 shows that the expected profit function π(p ,q ,r ) is expressed aswhere Π
_{G} (p ,q ,r ) and π_{L} (p ,q ,r ) are the π(p ,q ,r ) withβ _{2} =β _{2G} andβ _{2} =β _{2L}, respectively. The two functions π_{G} (p ,q ,r ) and π_{L} (p ,q ,r ) have a common point onp =r . Figure 10 illustrates an example of the contour of π(p ,q ,r ) with several values ofβ _{2G}. All of the functions π(p ,q ,r ) are concave in Figure 10. Whenβ _{2G} = 0 or 0.05, namely in the case of LA, the optimal price is equal to the reference pricer , otherwise π_{G} (p ,q ,r ) has the optimum.Similarly to the deterministic case, let
and
be the prices to maximize Π
_{G} (p ,q ,r ) and π_{L} (p ,q ,r ), respectively. A theorem is proved as a procedure to derive the optimal price for asymmetry consumers in the stochastic case.Theorem 4: When both the demand function D(p, r) and the inventory level Q are stochastic and consumers are LA or LS, the optimal price p^{*}(q, r) which maximizes the expected profit π(p, q, r )is derived by Equations (59) and (60) along with Equation (31): if Inequality (52) holds for p = p_{L}. Proof: Differentiating the left hand side of Inequality (52) with respect top shows that the left hand side monotonically increases with respect top . Inequality (52) hence holds for allp in the range [p_{L} ,p_{H} ] if it holds forp =p_{L} . Under the situation, Lemma 2 classifies the shape of Π(p ,q ,r ) in the same way as the deterministic case shown in Figures 6 and 7 for LA and LS consumers, respectively. Consequently, the optimal pricep ^{*}(q ,r ) is given by Equations (31), (59), and (60). □It is noticeable that Inequality (52) can be written as follows:
since
Inequality (61) has the same form as the wellknown critical fractile for newsvendor problems. It could be practical that the unit penalty cost
s for an opportunity loss is recognized as zero for daily perishable products. Inequality (61) always holds whens = 0.5. CONCLUSION
In this paper, an optimal clearance pricing in a single period has been discussed analytically considering consumer’ s reference price effect. The profit function in deterministic case is concave if target consumers are LN or LA. For LS consumers, the function is concave or bimodal. In the stochastic case, if Inequality (52) holds for
p =p _{L}, the contour of the expected profit function is similar to that of profit function in the deterministic case. The optimal price is obtained through the procedure proposed as Theorem 4. The optimal clearing price depends especially on consumers’ response, namely LN, LA, or LS.The resulting theorems in this paper can be applied to the optimal clearance pricing in a multiperiod case, which is the goal of our forthcoming study. The model in this paper can be modified to a combinatorial optimization, in which the firm determines the clearance price among several selectable prices, such as 10% off, 20% off, and 50% off. The modified model is more practical and the theorems in this paper could serve to reduce computational time to explore the optimal solutions.

[Figure 1.] π (p, q, r) for LN Consumers.

[Figure 2.] Regions for Optimal Price.

[Figure 3.] Optimal Prices in 3D Image.

[Figure 4.] πΛ(p, q, r) with Several Values of β2 on where r = 450.

[Figure 5.] p(q, r) for LA or LS Consumers.

[Figure 6.] πΛ(p, q, r) for LA Consumers.

[Figure 7.] πΛ(p, q, r) for LS Consumers.

[Figure 8.] π(p, q, r) with Several Values of β2 for LN Consumers where r = 450.

[Figure 9.] Fluctuation of π(p, q, r) with Respect to β2 for LN Consumers where r = 450.

[Figure 10.] π(p, q, r) with Several Values of β2G where β2L = 0.1 and r = 450.