BUS606: Operations and Supply Chain Management

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

$r_u(R) = (R−W)(b_u−aR)$,

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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

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

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

$u:[r_l(R),r_u(R)]→[0,1]$,

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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.

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

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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

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

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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.

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

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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".