Price-Setting Models for Innovative Products

In this section, the decision-making procedure for a retailer of innovative products is introduced. Before the selling season, the retailer must decide the order quantity q. For one unit of the innovative product, the wholesale price is $W$, and the retail price is $R$ and $R > W$. If there is an excess, the unit salvage price is $S_o > 0 (S_o < W)$. The unit opportunity cost is $S_u > 0$ for the shortage. The retailer's profit function is as follows:

$r(x,q)= \left\{ \begin{array}{ll} (R−W) x−(W−S_o)(q−x); x < q \\ (R−W)q−S_u(x−q); x ≥ q \end{array} \right.$,

(1)

The demand function of the innovative product is described by a random variable $X$ and the probability mass (density) function is $f(x)$.

Definition 1. Given the probability function $f(x)$ for random vector and $π:Ω→[0,1]$ be a function satisfying

$max{π(x):x∈Ω}=1$,

(2)
$π(x)$ is the relative likelihood degree  of $x$ if $π(x_1) > π(x_2)$ for $f(x_1) > f(x_2)$ and $π(x_1)=π(x_2)$ for $f(x_1)=f(x_2)$. For a retailer, his/her satisfaction levels towards profits are represented by the satisfaction function.

Definition 2. The satisfaction function is a strictly increasing function of the profit $r$,

$u : G → [0,1]$,

(3)

where $G$ is the set of profit $r$.

Obviously, the relative likelihood degrees and the satisfaction levels can be utilized to describe the relative position of the probabilities and the payoffs, respectively.

Usually, because of the short life cycle of the innovative product, there is only one chance given to the retailer to determine his/her order quantity and a unique demand will appear accordingly. Therefore, before ordering products, the retailer has to meditate which demand should be factored in. We take into account four types of demands (scenarios) for each order quantity with contemplating the likelihood degrees and the satisfaction levels, that is the demands with the higher satisfaction and likelihood (Type I), the lower satisfaction and higher likelihood (Type II), the higher satisfaction and lower likelihood (Type III), the lower satisfaction and likelihood (Type IV). It is intuitively acceptable that active, passive, daring and apprehensive retailers are inclined to take into account Type I, Type II, Type III and Type IV demands, respectively. Therefore, we call Type I, Type II, Type III and Type IV demands active, passive, daring and apprehensive focus points, respectively (shown in Table 1). Which kind of focus point is taking into account reflects the personality of the retailer under demand uncertainty.

Table 1. Four different focus points.

In the following model, a manufacturer who produces and sells an innovative product in the monopoly market is considered. Before selling season, the manufacturer produces $q$ units at unit cost W and setup cost is assumed to be zero. We consider the following widely used linear inverse demand function.

$x=b−aR$,

(15)

where $b > 0$ shows the limit demand when the retail price equals to $0$, and $a > 0$ represents the decreasing of demand when the retail price increases by one unit. We call a as the price sensitivity of market demand. The demand's uncertainty is described by the parameter b with probability density function $f(b)$. The profit function is with the retail price and the production quantity as the decision variables. With considering (1), it can be expressed as:

$r(R,b,q) = \left\{ \begin {array} {ll} (R−W)(b−aR)−(W−S_o)(q−b+aR); \quad b−aR < q \\ (R−W)q−S_u(b−aR−q); \quad b−aR ≥ q \end{array} \right.$

(16)

If one considers Definitions 1 and 2, then we have the relative likelihood function of $b$, i.e., $π(b)$ and the satisfaction function, i.e., $u(R,b,q)$. Similar to the newsvendor model, the following types of focus points are considered.

Active focus point: For retail price $R$ and production quantity $q$, the active focus point is

$b1(R,q) ∈ \overset{argmax}{x} lower[π(b),u(R,b,q)]$,

(17)

$b_1(R,q)−aR$ is the focused demand value with the relatively high likelihood degree and satisfaction level for the production quantity $q$.

Passive focus point: For retail price $R$ and production quantity $q$, the passive focus point is

$b_2(R,q) ∈ \overset{argmin}{x} upper[1−π(b),u(R,b,q)]$,

(18)

$b_2(R,q)−aR$ is the focused demand value with the relatively high likelihood degree and relatively low satisfaction level for the production quantity $q$.

Apprehensive focus point: For retail price $R$ and production quantity $q$, the apprehensive focus point is

$b_3 (R,q) ∈ \overset{argmin}{x} upper[π(b),u(R,b,q)]$,

(19)

$b_3(R,q)−aR$ is the focused demand value with the relatively low likelihood degree and satisfaction level for the production quantity $q$.

Daring focus point: For retail price $R$ and production quantity $q$, the daring focus point is

$b_4(R,q) ∈ \overset{argmin}{x}upper[π(b),1−u(R,b,q)]$,

(20)

$b_4(R,q)−aR$ is the focused demand value with the relatively low likelihood degree and relatively high satisfaction level for a production quantity $q$.

The sets of the four types of focus points of the retail price $R$ and production quantity $q$ are denoted as $B_1(R,q)$, $B_2(R,q)$, $B_3(R,q)$ and $B_4(R,q)$, respectively. The optimal production quantities for the manufacturers are

$q_1(R) ∈ \overset{argmax}{q} \overset{max}{b_1(R,q)  ∈ B_1(R,q)} \quad \quad u(R,b_1(R,q),q)$,

(21)

$q_2(R) ∈ \overset{argmax}{q} \overset{max}{b_2(R,q)  ∈ B_2(R,q)} \quad \quad u(R,b_2(R,q),q)$,

$q_3(R) ∈ \overset{argmax}{q} \overset{max}{b_3(R,q)  ∈ B_3(R,q)} \quad \quad u(R,b_3(R,q),q)$,

$q_4(R) ∈ \overset{argmax}{q} \overset{max}{b_4(R,q)  ∈ B_4(R,q)} \quad \quad u(R,b_4(R,q),q)$,

We suppose the following assumption in this section.

Assumption: The probability density function $f(b)$ is a unimodal function defined on the interval $[b_l,b_u]$, the mode is $bc  ∈ (b_l,b_u), f(b_l)=0$ and $f(b_u)=0$.

From Equation (15), we know $b_l−aR$ and $b_u−aR$ are the lowest demand and highest demand; $b_c−a_R$ is the most possible demand. Since the demand is within $[b_l−aR,b_u−aR]$, the reasonable supply should lie on the same interval. Therefore, the manufacturer's highest profit is

$r_u(R) = (R−W)(b_u−aR)$,

(29)

that is, the manufacturer produces the most $q=b_u−aR$, meanwhile, the demand happens to be the same as production quantity. The lowest profit is the minimum profit of two situations, one is the manufacturer produces the highest, however, the demand happens to be the least, the profit is $(b_l−aR) R+(b_u−b_l)S_o−(b_u−aR)W$; the other is the manufacturer produces the least, however, the demand happens to be the most, the profit is $(b_l−aR)(R−W)−(b_u−b_l)S_u$. Because of the high cost and margin of innovative products, it is reasonable to assume $W ≥ S_o + S_u$, which leads to

$r_l(R)=(b_l−aR)R+(b_u−b_l)S_o−(b_u−aR)W$.

(30)

For a fixed retail price $R$, the manufacturer's satisfaction level is the continuous strictly increasing function of profit $r$, that is

$u:[r_l(R),r_u(R)]→[0,1]$,

(31)

where $u(r_l(R))=0, u(r_u(R))=1$.

(31) provides a general formulation of the satisfaction function where the satisfaction level of the highest profit is 1, and the lowest profit is 0.

We have the following lemmas and propositions. The proofs are shown in the Appendix A. The following proposition indicates the relationships between the four types of manufacturers' focused profits.

Proposition 1. For any $R > W$, we have the following relationships between the four types of manufacturers' focused profits.

$r_4(R)>r_1(R)>r_2(R)>r_3(R)$.

(32)

Proposition 1 shows that the daring manufacturer always imagines a higher profit than the active manufacturer; meanwhile, the active manufacturer imagines a higher profit than a passive manufacturer, and the passive manufacturer imagines a higher profit than an apprehensive manufacturer. Such conclusions are interesting and intuitively acceptable.

Since the demand is not less than zero, it is reasonable that $R ∈ [W, \dfrac{b_l}{a}]$ and $b_l>aW$. In what follows, we suppose the satisfaction level is a linear function of profit, that is

$u(R,b,q) = \dfrac{r(R,b,q)−r_l(R)}{r_u(R)−r_l(R)}$,

(33)

The optimal retail price for four types of retailers is given in Propositions 2–5, below.

Proposition 2. If $∀b ∈ (b_c,b_u)$, $π(b)$ and $u(R,b,b−aR)$ are of class $C^1$, and $π′(b) ≠ \dfrac{∂u(R,b,b−aR)}{∂b}$ and $a > \dfrac{b_u−b_l}{(b_u−b_l)(S_o−W)π′(b)}$ hold, then the active profit function $r1(R)$ is concave. Furthermore, if $b_l−aW > b_u−b_l$, then the unique solution of $r′1(R)=0$ lies on the interval $R ∈ (W, \dfrac{b_l}{a})$, which is the optimal active retail price $R^{∗}_{1}$.

Proposition 2 shows that the active profit function's concavity is related to the price sensitivity of the market demand. Propositions 3–5 examine the concavities of passive, apprehensive and daring profit functions, respectively; and provide optimal passive, apprehensive and daring retail prices.

Proposition 3. If $π(b)$ is of class $C^1$ for $b  ∈ (b_l, b_c)$ and $b ∈ (b_c, b_u)$ and $u(R,b,q)$ is of  class $C^1$ for $q ∈ (b_l−aR,b_u−aR)$, $b ∈ (b_l, b_c)$ and $b ∈ (b_c, b_u)$ and $q′′_2 (R) > ξb′_2 (R,q_2 (R)) + ψb′′_2 (R,q_2(R)) + ζ$ (where $ξ=\dfrac{2}{W−S_o}$, $ψ = \dfrac{R−S_o}{W−S_o}$, $ζ=−\dfrac{2a}{W−S_o}$) holds, then the passive profit function $r_2(R)$ is concave. Furthermore, the unique solution of $r′_2(R) = 0$ lies on the interval $R ∈ (W, \dfrac{b_l}{a})$, which is the optimal passive retail price $R^{∗}_{2}$.

Proposition 3 points out that the passive profit function's concavity depends on the relationship between the changes in retail price $R$ of the optimal passive production quantity and of its corresponding focused demand value.

Proposition 4. The apprehensive profit function $r_3(R)$ is concave. Furthermore, if $b_l−aW > b_u−b_l$, then the optimal apprehensive retail price is the unique solution of $r′_3 (R) = 0$ within $R ∈ (W, \dfrac{b_l}{a})$.

Proposition 5. The daring profit function $r_4(R)$ is concave. If $b_l−aW > b_u−b_l$, then the optimal daring retail price is $R^{∗}_{4} =\dfrac{b_u +aW}{2a}$, and lies on the interval $R ∈ (W, \dfrac{b_l}{a})$; otherwise, $R^{∗}_{4}=\dfrac{b_l}{a}$.

Propositions 4 and 5 show that the apprehensive and daring profit functions $r_3(R)$ and $r_4(R)$ are always concave. Assume $r_1(R)$ and $r_2(R)$ are concave, the optimal retail prices lie on the interval $R ∈ (W, \dfrac{b_l}{a})$, we have Proposition 6 and 7, as follows.

Proposition 6. We have the following relationships between the four types of manufacturers' optimal retail prices.

$R^{∗}_{4} > R^{∗}_{1} > R^{∗}_{2} > R^{∗}_{3}$.

(34)

Proposition 6 tells that the daring manufacturer has the highest optimal retail price; the active manufacturer has a higher optimal price than the passive manufacturer, and the apprehensive manufacturer has the lowest optimal retail price. Such conclusions can be used for distinguishing the type of the manufacturer according to the observed retail price which he/she has set and also can predict the retail price which he/she will set if knowing the personality of the manufacturer. Let us look at the supporting evident form the Wall Street Journal which reported that the manufacturers who sell ultraluxury brands are actually risk-takers, they are raising prices to distinguish their products from other luxury goods, and they believe that the rich consumers are willing to accept such prices.

Proposition 7. The optimal active, passive, apprehensive and daring retail prices are decreasing in $a$.

Proposition 7 shows that the with the increase of the market demand's price sensitivity, the manufacturer will charge a lower retail price; that is to say, whichever the manufacturer's type is, decreasing the price sensitivity of the demand is efficient for charging a high retail price in the innovative product market. Interestingly, the following fact supports the above conclusion. It is from the report of Accenture® that the luxury manufacturers that build brands on the image and lifestyle are able to withstand bigger competitive pricing differences than manufacturers who build their brands on the price. It future explained that "a well-known luxury manufacturer incorporated the price sensitivity metrics into its overall pricing and assortment strategy in recent years. The strategy has helped boost the company's profit margins to its highest level".

A direct-sale store of a fashion company from France, located in Dalian, China, is going to sell a new design fashion clothes. The fashion store is a monopoly in the northeast China market. The unit cost $W$, salvage price $S_o$ and opportunity cost $S_u$ are 7000, 1000 and 4000 (RMB), respectively. The market demand is related to the retail price, and we have $b_l=1000$, $b_u=1500$. Let us consider the store's pricing policies when $a=0.02$, $a=0.05$ and $a=0.10$. As an example, let us see the details when $a=0.05$.

Suppose parameter $b$'s probability density function is $f(b)=0.004−|\dfrac{b−1250}{62500}|$. From Equation (15), the profit function is

$r(R,b,q)= \left\{ \begin{array} {ll} (R−7000)(b−0.05R)−6000(q−b+0.05R); b−0.05R < q \\ (R−7000)q−4000(b−0.05R−q); \quad b−0.05R  ≥ q \end{array} \right.$

(35)

For simplification, the satisfaction function is the normalization of $r(R,b,q)$. We obtain $R^{∗}_{1}=16,767$, $R^{∗}_{2}=14,328$, $R^{∗}_{3}=13,919$ and $R^{∗}_{4}=18,500$; $r^{∗}-{1}=459,800$, $r^{∗}_{2}=2,792,900$, $r^{∗}_{3}=1,394,500$ and $r^{∗}_{4}=6,612,500$. Similarly, we can obtain that when $a=0.02$, $R^{∗}_{1}=36,804$, $R^{∗}_{2}=29,384$, $R^{∗}_{3}=28,797$, $R^{∗}_{4}=41,000$, $r^{∗}_{1}=17,507,000$, $r^{∗}_{2}=10,766,000$, $r^{∗}_{3}=8,865,800$ and $r^{∗}_{4}=23,120,000$; when $a=0.10$, $R^{∗}_{1}=10,000$, $R^{∗}_{2}=9033$, $R^{∗}_{3}=8922$, $R^{∗}_{4}=10,000$, $r^{∗}_{1}=857,140$, $r^{∗}_{2}=400,110$, $r^{∗}_{3}=−799,350$ and $R^{∗}_{4}=1,500,000$. The relationships between retail prices and profits when $a=0.02$, $a=0.05$ and $a=0.10$ are shown in Figure 1, Figure 2 and Figure 3, respectively.

Figure 1. Relationships between retail prices and profits when $a=0.02$.

Figure 2. Relationships between retail prices and profits when $a=0.05$.

Figure 3. Relationships between retail prices and profits when $a=0.10$.

The numerical example shows three interesting phenomena. First, we observe that with the increasing of market demand's price sensitivity (the increase of parameter $a$), all of the four types of manufacturers charge the lower retail prices. Second, the numerical example indicates that for any $R > W$ , $r_4(R) > r_1 (R) > r_2 (R) > r_3(R)$. That is the focused profits of the daring manufacturer are higher than the ones of active one; the focused profits of the active manufacturer are higher than the ones of the passive manufacturer; the focused profits of the passive manufacturer are higher than the ones of the apprehensive manufacturer. Third, we have $R^{∗}_{4}  > R^{∗}_{1} > R^{∗}_{2} > R^{∗}_{3}$  which shows that the daring manufacturer has the highest optimal retail price; the active manufacturer has a higher optimal price than the passive manufacturer, and the apprehensive manufacturer has the lowest optimal retail price. The first result is similar to Reference, however, since we model the behaviors of different types of manufacturers, the second and third results are original. Such conclusions are in accordance with phenomena in the real business world.