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.

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 p_H. 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 [p_L, p_H]. 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 [p_L, p_H]. The inventory level Q is assumed to be a random variable and it is given by Q = q + ε_q, where q is the average of Q and ε_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 and β_2L 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 β_2G < β_2L, β_2G = β_2L, and β_2G> β_2L are called LA, LN, and LS, respectively. All of the parameters B_0G(r), B_0L(r), B_1G, B_1L are consequently positive. The random factor ε_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 price p and reference price r. Assume ?h < c < p_L and D(p, r) > 0 for any p, r ∈ [P_L, P_H].

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 and ε_q are constantly equal to 0.

This subsection confines our discussion to the optimal pricing for LN consumers. Let β_2G = β_2L = β₂ then the demand function D(p, r) = d(p, r) is given by the following equation:

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

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

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

Let

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

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:

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

Meanwhile, 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 of

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

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

and

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

and 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 gives

Hence, both

and

increase monotonically with respect to r and the increment of

is 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 three-dimensional images in Figure 3 under the following parameters setting: β₀ = 100, β₁ = 0.1, β₂ = 0.01, and h = 50.

Here, let

be the β₂ which satisfies

then Equation (10) yields

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

Lemma 1: The price

is decreasing with respect to β₂. The profit π_Λ(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 π_Λ(p, q, r) is decreasing and increasing with respect to β₂ for

and

respectively.

Proof: Differentiating Equation (10) derives

hence

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

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

with respect to β₂ gives

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

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

Since

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

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

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 function d(p, r) can be expressed as follows:

since both d_G(p, r) and d_L(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

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) for

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

where π_ΘG(p, q, r) and π_ΘL(p, q, r) are the π_Θ(p, q, r) in Equation (8) with β₂ = β_2G and β₂ = β_2L, respectively. The function π_Θ(p, q, r) in Equation (26) monotonically increases with respect to p since both π_ΘG(p, q, r) and π_ΘL(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

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

since Equation (9) shows π_Λ(p, q, r) increases as d(p, r) increases. The profit functions π_ΛG(p, q, r) and π_ΛL(p, q, r) are the π_Λ(p, q, r) in Equation (9) with β₂ = β_2G and β₂ = β_2L, respectively. The two profit functions π_ΛG(p, q, r) and π_ΛL(p, q, r) have a common point on p = r. Let

and

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

Lemma 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 of

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:

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

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

and

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

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

In the other cases, the cardinality is one and Equation (32) explores the optimal price without Equation (30).

This section discusses the optimal pricing in case that the demand D(p, r) and inventory level Q are stochastic.

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 = ε_q-ε_d. in accordance with Petruzzi and Dada (1999). Note that

The average of ε is 0 and the range of ε is [q_L - d_H, q_H - d_L]. Then, the profit π(p, q, r) is expressed

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

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

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

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:

where

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

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

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

which 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 β₂ which satisfies

then

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 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

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

respectively; (iii)

is decreasing with respect to β₂.

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

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 π_Λ(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.

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 as

where Π_G(p, q, r) and π_L(p, q, r) are the π(p, q, r) with β₂ = β_2G and β₂ = β_2L, respectively. The two functions π_G(p, q, r) and π_L(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 β_2G. All of the functions π(p, q, r) are concave in Figure 10. When β_2G = 0 or 0.05, namely in the case of LA, the optimal price is equal to the reference price r, 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 to p shows that the left hand side monotonically increases with respect to p. Inequality (52) hence holds for all p in the range [p_L, p_H] if it holds for p = 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 price p^*(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 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.

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 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.