Price-Setting Models for Innovative Products

Price-Setting Newsvendor Models Based on OSDT

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)

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

(22)

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

(23)

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

(24)

From Equations (17)–(24), we can see that for a fixed

$R$ , the profit functions of the active, passive, apprehensive and daring manufacturers are

$r(R,b_1(R,q_1(R)),q_1(R))$ ,

$r(R,b_2(R,q_2(R)),q_2(R))$ ,

$r(R,b_3(R,q_3(R)),q_3(R))$ and

$r(R,b_4(R,q_4(R)),q_4(R))$ , respectively, which are named as active, passive, apprehensive and daring profit functions, respectively. Because they are the functions of single variable

$R$ , for simplicity, we use

$r_1(R)$ ,

$r_2(R)$ ,

$r_3(R)$ and

$r_4(R)$ in the following parts. For each type of manufacturer, the optimal retail price is which to maximize his/her profit function.

$R^{∗}{1} ∈ \overset{argmaxr1}{R} (R), q^{∗}_{1} ∈ q^1 (R^{∗}_{1})$ ,

(25)

$R^{∗}{2} ∈ \overset{argmax r_2}{R} (R), q^{∗}_{2} ∈ q^2 (R^{∗}_{2})$ ,

(26)

$R^{∗}{3} ∈ \overset{argmax r_3}{R} (R), q^{∗}_{3} ∈ q^3 (R^{∗}_{3})$ ,

(27)

$R^{∗}{4} ∈ \overset{argmax r_4}{R} (R), q^{∗}_{4} ∈ q^4 (R^{∗}_{4})$ ,

(28)

$R^{∗}_{1}$ ,

$R^{∗}_{2}$ ,

$R^{∗}_{3}$ and

$R^{∗}_{4}$ are optimal retailer prices for active, passive, apprehensive and daring manufacturers, respectively. They are named as optimal active, passive, apprehensive and daring retail prices, respectively.