A Mathematical Analysis on Daily Inventory Clearance Pricing with Consumer’s Reference Price

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  • 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 long-term 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 loss-neutral. A sufficient condition is shown to obtain an optimal price for loss-averse and loss-seeking 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 long-term 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 well-researched 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 long-term 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 well-known 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 loss-neutral (LN), loss-averse (LA), or loss-seeking (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 price pH. No replenishment is considered in this model. After closing time, unsold products are salvaged or disposed at a unit cost h, which means that the unsold products are salvaged if h < 0 and they are disposed otherwise. Let s (> 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 [pL, pH]. The firm determines the price p before the prescheduled time in advance with considering uncertain inventory level Q and supposed consumer’s reference price r for the products. If some products are unsold at the prescheduled time, the unsold products are sold at price p from then to closing time. The reference price r exists in the range [pL, pH]. The inventory level Q is assumed to be a random variable and it is given by Q = q + εq, where q is the average of Q and εq is a random factor whose mean is 0 and range is [qL, qH].

    The demand function for the product D(p, r) includes the reference price effect and stochastic variation:

    image
    image
    image
    image

    The positive parameters β2G and β2L respectively represent the degree of the reference price effect when consumers recognize the discount price p as a gain (p < r) and a loss (p > r). The parameters characterize the consumer’s response toward the selling price. The consumers with β2G < β2L, β2G = β2L, and β2G> β2L are called LA, LN, and LS, respectively. All of the parameters B0G(r), B0L(r), B1G, B1L are consequently positive. The random factor εd in the demand function, whose mean is 0 and range is [dL, dH], is assumed to be independent of both the sales price p and reference price r. Assume ?h < c < pL and D(p, r) > 0 for any p, r ∈ [PL, PH].

    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

    εd and εq are constantly equal to 0.

       3.1 Optimal Pricing for Loss-Neutral Consumers

    This subsection confines our discussion to the optimal pricing for LN consumers. Let β2G = β2L = β2 then the demand function D(p, r) = d(p, r) is given by the following equation:

    image

    The both parameters B0(r) and B1 are positive. Let

    image

    be the price p which satisfies d(p, r) = q. Then, it holds

    image

    When the products are sold at price p for lossneutral consumers with reference price r and the inventory level Q is a constant q, the profit π(p, q, r) is represented by

    image
    image
    image

    Let

    image

    be the price p which maximizes πΛ(p, q, r). A simple analysis of πΛ(p, q, r) derives

    image

    Let p* (q, r) be the price to maximize π(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:

    image

    Proof: The function πΘ(p, q, r) monotonically increases since

    image

    Meanwhile, the function πΛ(p, q, r) is a concave function shown as follows:

    image
    image

    Hence, the profit π (p, q, r) increases monotonically in the range of

    image

    where π(p, q, r) = πΘ(p, q, r), and it is concave in the range of

    image

    where π(p, q, r) = πΛ(p, q, r). Figure 1 depicts the shape of π(p, q, r). When the two prices

    image

    and

    image

    exits in [pL, pH], the profit function π(p, q, r) monotonically increases for

    image

    and monotonically decreases for

    image

    Then, the greater between

    image

    and

    image

    maximizes the profit π(p, q, r). □

    Corollary 1: The optimal price p* (q, r) by Equation (11) is expressed as follows:

    image

    Proof:

    image
    image
    image

    The above inequalities directly derive the corollary. □

    Differentiating

    image

    and

    image

    with respect to r gives

    image

    Hence, both

    image

    and

    image

    increase monotonically with respect to r and the increment of

    image

    is twice that of

    image

    Figure 2 illustrates the region regarding the optimal price with assuming

    image

    and

    image

    An numerical example for the optimal price p* (q, r) are shown as three-dimensional images in Figure 3 under the following parameters setting: β0 = 100, β1 = 0.1, β2 = 0.01, and h = 50.

    Here, let

    image

    be the β2 which satisfies

    image

    then Equation (10) yields

    image

    Lemma 1 explains the influence of β2 on πΛ(p, q, r) and

    image

    Lemma 1: The price

    image

    is decreasing with respect to β2. The profit πΛ(p, q, r) is increasing, constant, and decreasing with respect to β2 for p < r, p = r, and p > r, respectively. Furthermore, the maximum of πΛ(p, q, r) is decreasing and increasing with respect to β2 for

    image

    and

    image

    respectively.

    Proof: Differentiating Equation (10) derives

    image

    hence

    image

    is decreasing with respect to β2. Similarly, differentiating Equation (9) derives

    image

    From the assumption that p+h > 0, Equation (22) proves the property on πΛ(p, q, r). Finally, differentiating

    image

    with respect to β2 gives

    image

    Equations (21), (22), and (23) prove the last property regarding the maximum of profit

    image

    The profit functions πΛ(p, q, r) with several values of β2 are depicted in Figure 4, where r = 450 and

    image

    Since

    image

    decreases monotonically with respect to β2, the maximum vertex of πΛ(p, q, r) moves leftward and downward and approaches toward the point (r, πΛ (r, q, r)) with increasing β2 to

    image

    The vertex of πΛ(p, q, r) moves leftward and upward from (r, πΛ (r, q, r)) when β2 increases from

    image

       3.2 Optimal Pricing for Asymmetry Consumers

    This subsection discusses the optimal pricing for LA and LS consumers, namely when it holds that β2Gβ2L. In this case, the expected demand function d(p, r) can be expressed as follows:

    image

    since both dG(p, r) and dL(p, r) linear functions with respect to p and they have a common point at p = r.

    In the same manner in the previous subsection, as

    image

    was defined as follows:

    image

    Figure 5 indicates the two prices

    image

    and

    image

    Since Equation (7) also holds for LA and LS consumers, π(p, q, r) = πΘ(p, q, r) for

    image

    From Equation (8), the profit function πΘ(p, q, r) in the case of inventory shortage is decreasing as d(p, r) increases, and then it is expressed as

    image

    where πΘG(p, q, r) and πΘL(p, q, r) are the πΘ(p, q, r) in Equation (8) with β2 = β2G and β2 = β2L, respectively. The function πΘ(p, q, r) in Equation (26) monotonically increases with respect to p since both πΘG(p, q, r) and πΘL(p, q, r) increase monotonically as proved by Equation (12). The profit π(p, q, r) for LA and LS consumers increases monotonically in the range

    image

    The profit function πΛ(p, q, r) in the case of excessive inventory is expressed as

    image

    since Equation (9) shows πΛ(p, q, r) increases as d(p, r) increases. The profit functions πΛG(p, q, r) and πΛL(p, q, r) are the πΛ(p, q, r) in Equation (9) with β2 = β2G and β2 = β2L, respectively. The two profit functions πΛG(p, q, r) and πΛL(p, q, r) have a common point on p = r. Let

    image

    and

    image

    be respectively the prices on which πΛG(p, q, r) and πΛL(p, q, r) have a maximum, namely

    image
    image

    Lemma 1 concludes that it holds

    image

    LA consumers and

    image

    for LS consumers. Then, Lemma 1 restricts the possibility of the shapes of the profit functions πΛ(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 of

    image

    when 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:

    image
    image
    image

    Proof: The set P1* consists of the candidate prices to maximize πΛ(p, q, r) without considering the lower and upper limits of the sales price. For

    image

    in the middle case in Figure 7, πΛ(p, q, r) is bimodal and both

    image

    and

    image

    could be optimal. In the case of max

    image

    is the optimal price both for LA and LS consumers. Similarly,

    image

    is the optimal

    when it holds min

    image

    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 in P1* to be optimal for π(p, q, r). □

    Note that the cardinality of P2* is two in the case of

    image

    In 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 level Q 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 variables z = q-d(p, r) and ε = εq-εd. in accordance with Petruzzi and Dada (1999). Note that

    image

    The average of ε is 0 and the range of ε is [qL - dH, qH - dL]. Then, the profit π(p, q, r) is expressed

    image

    Let f ( ? ) and F( ? ) be the probabilistic density function and the distribution function of the variable ε. Define

    image

    The expected profit Π(p, q, r), hence, is obtained by

    image

    The expected profit Π(p, q, r) can be rewritten by

    image
    image
    image
    image
    image

    In Equation (36), Ψ(p, r) and L(p, z) respectively imply the profit for Q = 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 price p*(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:

    image

    where

    image

    is the unique solution of g(p, q, r) = 0:

    image

    Proof: Differentiating from Equations (36) to (40) with respect to pyields

    image
    image
    image
    image
    image
    image
    image

    Using the assumption of h < pL, Equations (48) and (49) prove that π(p, q, r) is concave with respect to p and has a unique maximum

    image

    which satisfies g(p, q, r) = 0. □

    Corollary 2: The optimal price p* (q, r) by Equation (41) is expressed as follows:

    image

    Proof: It is obviously proved from the concavity of the function π(p, q, r). □

    Similarly to the deterministic case, let

    image

    be the β2 which satisfies

    image

    then

    image

    is obtained from the equation g(r, q, r) = 0:

    image

    provided that

    image

    is not equal to 0. The next lemma reveals a property on Π (p, q, r) with respect to β2.

    Lemma 2: The expected profit Π(p, q, r) is concave with respect to β2. Furthermore, under the circumstance which satisfies the following inequality

    image

    Π(p, q, r) has the following properties: (i) Π (p, q, r) is increasing, constant, and decreasing with respect to β2 for p < r, p = r, and p > r, respectively; (ii) The maximum of Π (p, q, r) is decreasing and increasing with respect to β2 for

    image

    respectively; (iii)

    image

    is decreasing with respect to β2.

    Proof: Differentiating from Equations (36) to (40) with respect to β2 yields

    image
    image
    image
    image
    image

    Equation (57) proves the concavity of Π(p, q, r). Equation (56) proves property (i) from the assumption that p + 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 of β2. □

    Lemma 2 indicates that the contour of the expected profit function Π(p, q, r) shown in Figure 8 which resembles that of πΛ(p, q, r) shown in Figure 4. Figure 9 depicts the fluctuation of π(p, q, r) with respect to β2 for p = 410, 430, 450, 470, and 490 where r = 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 case p = 410. The functionr π(p, q, r) with p = 410 decreases for β2 > 0.2, where Inequality (52) does not hold.

       4.2 Optimal Pricing for Asymmetry Consumers

    In the case of β2Gβ2L , if Inequality (52) holds, Lemma 2 shows that the expected profit function π(p, q, r) is expressed as

    image

    where ΠG(p, q, r) and πL(p, q, r) are the π(p, q, r) with β2 = β2G and β2 = β2L, respectively. The two functions πG(p, q, r) and πL(p, q, r) have a common point on p = r. Figure 10 illustrates an example of the contour of π(p, q, r) with several values of β2G. All of the functions π(p, q, r) are concave in Figure 10. When β2G = 0 or 0.05, namely in the case of LA, the optimal price is equal to the reference price r, otherwise πG(p, q, r) has the optimum.

    Similarly to the deterministic case, let

    image

    and

    image

    be the prices to maximize ΠG(p, q, r) and πL(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):

    image
    image

    if Inequality (52) holds for p = pL.

    Proof: Differentiating the left hand side of Inequality (52) with respect to p shows that the left hand side monotonically increases with respect to p. Inequality (52) hence holds for all p in the range [pL, pH] if it holds for p = pL. 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 price p*(q, r) is given by Equations (31), (59), and (60). □

    It is noticeable that Inequality (52) can be written as follows:

    image

    since

    image

    Inequality (61) has the same form as the well-known 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 when s = 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 = pL, 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 multi-period 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.

  • 1. Arrow K., Harris T., Marschak J. (1951) Optimal inventory policy [Econometrika] Vol.19 P.250-272 google doi
  • 2. Greenleaf E. A. (1995) The Impact of Reference Price Effects on the Profitability of Price Promotions [Marketing Science] Vol.14 P.82-104 google doi
  • 3. Kahneman D., Tversky A. (1979) Prospect theory: An analysis of decision under risk [Econometrica] Vol.47 P.263-291 google doi
  • 4. Kalyanaram G., Winer R. S. (1995) Empirical Generalization from Reference Price Research [Marketing Science] Vol.14 P.G161-169 google doi
  • 5. Koide T., Sandoh H. (2009) Reference Price Effect and Inventory Constraint on Optimal Pricing for Daily Perishable Products [Proceedings on the 2009 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM)] P.370-374 google
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  • [Figure 1.] π (p, q, r) for LN Consumers.
    π (p, q, r) for LN Consumers.
  • [Figure 2.] Regions for Optimal Price.
    Regions for Optimal Price.
  • [Figure 3.] Optimal Prices in 3D Image.
    Optimal Prices in 3D Image.
  • [Figure 4.] πΛ(p, q, r) with Several Values of β2 on where r = 450.
    πΛ(p, q, r) with Several Values of β2 on where r = 450.
  • [Figure 5.] p(q, r) for LA or LS Consumers.
    p(q, r) for LA or LS Consumers.
  • [Figure 6.] πΛ(p, q, r) for LA Consumers.
    πΛ(p, q, r) for LA Consumers.
  • [Figure 7.] πΛ(p, q, r) for LS Consumers.
    πΛ(p, q, r) for LS Consumers.
  • [Figure 8.] π(p, q, r) with Several Values of β2 for LN Consumers where r = 450.
    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.
    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.
    π(p, q, r) with Several Values of β2G where β2L = 0.1 and r = 450.