Price-Setting Models for Innovative Products

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.