Joint optimal pricing and advertising policies in a fashion supply chain under the ODM strategy considering fashion level and goodwill

Abstract

The original design manufacturing (ODM) strategy has become popular in the fashion supply chain field. We consider apparel’s fashion level and advertising effort as important influential factors on goodwill, and present a differential game involving a fashion brand and a supplier. The supplier controls the design improvement effort applied to the apparel and sells it to end consumers through the fashion brand, which controls the advertising effort and retail price. In the case in which the demand for products is affected by goodwill, retail price and promotion, centralised and decentralised differential game models are constructed. The dynamic wholesale price contract is then introduced to solve the external coordination problem. Furthermore, a sensitivity analysis of the related parameters is conducted using the numerical simulation method. It is found that the introduction of the dynamic wholesale price contract increases design investment, promotion effort, and goodwill towards and demand for the products. We also derive the feasible region of wholesale prices in which both members are willing to implement the commitment scenario. The win–win region becomes larger as the effectiveness of design innovation increases, whereas it becomes smaller as price sensitivity increases.

Appendices

Appendix A

Proof of Proposition 1

The optimization problem of the fashion supply chain is given as

$$\mathop {\hbox{max} }\limits_{I\left( t \right),p\left( t \right),A\left( t \right)} J_{SC}^{C} = \int_{0}^{\infty } {e^{ - \rho t} \left\{ {p\left( t \right)Q\left( t \right) - C_{I} \left( t \right) - C_{A} \left( t \right)} \right\}dt} .$$

(A.1)

Let the value function $V_{SC}^{C} \left( {\theta ,G} \right) = \mathop {\hbox{max} }\limits_{I,p,A} \int_{t}^{\infty } {e^{ - \rho (\tau - t)} } \left\{ {p\left( t \right)Q\left( t \right) - C_{I} \left( t \right) - C_{A} \left( t \right)} \right\}d\tau$; the optimal profit function of the fashion supply chain then is given as

$$J_{SC}^{C} = e^{ - \rho t} V_{SC}^{C} \left( {\theta ,G} \right).$$

(A.2)

The value function of the fashion supply chain $V_{SC}^{C} \left( {\theta ,G} \right)$ satisfies the HJB equation as

$$\rho V_{SC}^{C} \left( {\theta ,G} \right) = \mathop {\hbox{max} }\limits_{I,p,A} \left\{ \begin{aligned} p\left( {a - bp} \right)\left( {\mu \theta + \phi G} \right) - \frac{1}{2}k_{1} I^{2} - \frac{1}{2}k_{2} A^{2} \hfill \\ + V_{SC\theta }^{C'} \left( {\delta I - \eta \theta } \right) + V_{SCG}^{C'} \left( {\tau A + r\theta - \varepsilon G} \right) \hfill \\ \end{aligned} \right\},$$

(A.3)

Maximization of the right-hand side of the HJB formula with respect to I, p and A yields

$$I = \frac{{\delta V_{SC\theta }^{C'} }}{{k_{1} }},p = \frac{a}{2b},A = \frac{{\tau V_{SCG}^{C'} }}{{k_{2} }},$$

(A.4)

Inserting (A.2) and (A.3) on the right-hand side of the HJB equation provides

$$\rho V_{SC}^{C} \left( {\theta ,G} \right) = \left( {\frac{{\mu a^{2} }}{4b} - \eta V_{SC\theta }^{C'} + rV_{SCG}^{C'} } \right)\theta + \left( {\frac{{\phi a^{2} }}{4b} - \varepsilon V_{SCG}^{C'} } \right)G + \frac{{\left( {\delta V_{SC\theta }^{C'} } \right)^{2} }}{{2k_{1} }} + \frac{{\left( {\tau V_{SCG}^{C'} } \right)^{2} }}{{2k_{2} }},$$

(A.5)

Note that $V_{SC}^{C} \left( {\theta ,G} \right)$ is the value function for the maximization problem P1. Guided by the model’s linear structure, we conjecture that the fashion supply chain’ value function is linear and given by

$$V_{SC}^{C} \left( {\theta ,G} \right) = m_{1} \theta + m_{2} G + m_{3} , ,$$

(A.6)

and follows from (A.6) that

$$V_{SC\theta }^{C'} = m_{1} ,\quad V_{SCG}^{C'} = m_{2} ,$$

(A.7)

Substituting (A.6) and (A.7) into (A.5), and then equating the coefficients of θ, G on both sides of (A.5), we get the expressions of m₁, m₂, and m₃ given by:

$$\left\{ {\begin{array}{*{20}l} {m_{1}^{*} = \frac{{a^{2} }}{{4b\left( {\rho + \eta } \right)}}\left( {\mu + \frac{\phi r}{\rho + \varepsilon }} \right)} \hfill \\ {m_{2}^{*} = \frac{{\phi a^{2} }}{{4b\left( {\rho + \varepsilon } \right)}}} \hfill \\ {m_{3}^{*} = \frac{{\delta^{2} m_{1}^{*2} }}{{2\rho k_{1} }} + \frac{{\tau^{2} m_{2}^{*2} }}{{2\rho k_{2} }}} \hfill \\ \end{array} } \right.,$$

(A.8)

Substituting (A.8) into (A.4), we get the expressions of the optimal selling price, advertising and design innovation investment strategies shown in Proposition 1.

Substituting optimal strategies (10) into (1) and (2), we can obtain the following homogeneous equations,

$$\left[ {\begin{array}{*{20}c} {\mathop \theta \limits^{ \bullet } } \\ {\mathop G\limits^{ \bullet } } \\ \end{array} } \right] = \bar{B}\left[ {\begin{array}{*{20}c} \theta \\ G \\ \end{array} } \right] + \beta ,$$

(A.9)

where $\bar{B} = \left[ {\begin{array}{*{20}c} { - \eta } & 0 \\ r & { - \varepsilon } \\ \end{array} } \right]$, $\beta = \left[ {\begin{array}{*{20}c} {\delta E^{C*} } \\ {\varepsilon A^{C*} } \\ \end{array} } \right]$. The characteristic roots of formula (A.9) are

$$\lambda_{1} = - \eta ,\quad \lambda_{2} = - \varepsilon$$

(A.10)

The characteristic vectors of formula (A.9) are

$$H = \left[ {\begin{array}{*{20}l} {\left( {\eta - \varepsilon } \right)/r} \hfill & 0 \hfill \\ 1 \hfill & 1 \hfill \\ \end{array} } \right],$$

(A.11)

Substituting (A.11) and (A.10) into (A.9), we have,

$$\left[ {\begin{array}{*{20}c} {\theta \left( t \right)} \\ {G\left( t \right)} \\ \end{array} } \right] = H\left[ {\begin{array}{*{20}l} {e^{ - \eta t} } \hfill & 0 \hfill \\ 0 \hfill & {e^{ - \varepsilon t} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {c_{1} } \\ {c_{2} } \\ \end{array} } \right] - \bar{B}^{ - 1} \beta ,$$

(A.12)

where c₁ and c₂ are constants to be determined.

Since the initial conditions $\theta \left( 0 \right) = \theta_{0}$ and $G\left( 0 \right) = G_{0}$, we have,

$$c_{1} = \frac{{Q_{0} r}}{\eta - \varepsilon } - \frac{{E^{C*} \delta r}}{{\eta \left( {\eta - \varepsilon } \right)}},$$

(A.13)

$$c_{2} = G_{0} - G_{\infty } - c_{1} ,$$

(A.14)

Therefore, the general solutions of (A.12) are given by

$$\left\{ {\begin{array}{*{20}l} {\theta^{C*} \left( t \right) = \left( {\theta_{0} - \theta_{\infty }^{C} } \right)e^{ - \eta t} + \theta_{\infty }^{C} } \hfill \\ {G^{C*} \left( t \right) = \frac{r}{\varepsilon - \eta }\left( {\theta_{0} - \theta_{\infty }^{C} } \right)e^{ - \eta t} + \left( {G_{0} - G_{\infty }^{C} - \frac{r}{\varepsilon - \eta }\left( {\theta_{0} - \theta_{\infty }^{C} } \right)} \right)e^{ - \varepsilon t} + G_{\infty }^{C} } \hfill \\ \end{array} } \right.,$$

(A.15)

where $\theta_{\infty }^{C}$ and $G_{\infty }^{C}$ are the particular solution to system (A.12) shown in Proposition 2.This completes the proof. □

Appendix B

Proof of Proposition 2

Because the game is played à la Stackelberg and the fashion brand is the leader, we first derive the decision variables for the supplier for the second game stage. The optimization problem of supplier is given as

$$\mathop {\hbox{max} }\limits_{I\left( t \right)} J_{S}^{D} = \int_{0}^{\infty } {e^{ - \rho t} \left\{ {\omega Q\left( t \right) - C_{I} \left( t \right)} \right\}dt} .$$

(B.1)

Let the value function $V_{S}^{D} \left( {\theta ,G} \right) = \mathop {\hbox{max} }\limits_{I} \int_{t}^{\infty } {e^{ - \rho (\tau - t)} } \left\{ {\omega Q\left( t \right) - C_{I} \left( t \right)} \right\}d\tau$; the optimal profit function of supplier then is given as

$$J_{S}^{D} = e^{ - \rho t} V_{S}^{D} \left( {\theta ,G} \right).$$

(B.2)

Similarly, let $V_{B}^{D} \left( {\theta ,G} \right)$ denote value function of the brand; the optimal profit function of the fashion brand is given by

$$J_{B}^{D} = e^{ - \rho t} V_{B}^{D} \left( {\theta ,G} \right). .$$

(B.3)

The supplier’s HJB is

$$\rho V_{S}^{D} \left( {\theta ,G} \right) = \mathop {\hbox{max} }\limits_{I} \left\{ \begin{aligned} \omega \left( {a - bp} \right)\left( {\mu \theta + \phi G} \right) - \frac{1}{2}k_{1} I^{2} + \hfill \\ V_{S\theta }^{D'} \left( {\delta I - \eta \theta } \right) + V_{SG}^{D'} \left( {\tau A + r\theta - \varepsilon G} \right) \hfill \\ \end{aligned} \right\},$$

(B.4)

and its maximization provides the necessary condition for design innovation,

$$I = \frac{{\delta V_{S\theta }^{D'} }}{{k_{1} }}.$$

(B.5)

Similarly, the fashion brand’s HJB equation is

$$\rho V_{B}^{D} \left( {\theta ,G} \right) = \mathop {\hbox{max} }\limits_{p,A} \left\{ \begin{aligned} \left( {p - \omega } \right)\left( {a - bp} \right)\left( {\mu \theta + \phi G} \right) - \frac{1}{2}k_{2} A^{2} \hfill \\ + V_{B\theta }^{D'} \left( {\delta I - \eta \theta } \right) + V_{BG}^{D'} \left( {\tau A + r\theta - \varepsilon G} \right) \hfill \\ \end{aligned} \right\},$$

(B.6)

and its maximisation provides the necessary condition for selling price and advertising effort

$$p = \frac{a + b\omega }{2b},A = \frac{{\tau V_{BG}^{D'} }}{{k_{2} }}. .$$

(B.7)

By inserting (B.5) and (B.7) inside the HJB equations, we obtain the following two algebraic equations,

$$\begin{aligned} \rho V_{S}^{D} \left( {\theta ,G} \right) & = \left( {\frac{{\omega \mu \left( {a - b\omega } \right)}}{2} - \eta V_{S\theta }^{D'} + rV_{SG}^{D'} } \right)\theta + \left( {\omega \phi \left( {a - b\omega } \right) - \varepsilon V_{SG}^{D'} } \right)G \\ & \quad + \,\frac{{\tau^{2} V_{BG}^{D'} V_{SG}^{D'} }}{{k_{2} }} + \frac{{\delta^{2} \left( {V_{S\theta }^{D'} } \right)^{2} }}{{2k_{1} }}, \\ \end{aligned}$$

(B.8)

$$\begin{aligned} \rho V_{B}^{D} \left( {\theta ,G} \right) & = \left( {\frac{{\mu \left( {a - b\omega } \right)^{2} }}{4b} - \eta V_{B\theta }^{D'} + rV_{BG}^{D'} } \right)\theta + \left( {\frac{{\phi \left( {a - b\omega } \right)^{2} }}{4b} - \varepsilon V_{BG}^{D'} } \right)G \\ & \quad + \,\frac{{\delta^{2} V_{B\theta }^{D'} V_{S\theta }^{D'} }}{{k_{1} }} + \frac{{\tau^{2} \left( {V_{BG}^{D'} } \right)^{2} }}{{2k_{2} }}, \\ \end{aligned}$$

(B.9)

Thus, we obtain the following linear forms for the value functions,

$$V_{S}^{D} \left( {\theta ,G} \right) = l_{1} \theta + l_{2} G + l_{3} ,\quad V_{B}^{D} \left( {\theta ,G} \right) = n_{1} \theta + n_{2} G + n_{3} ,$$

(B.10)

in which l₁, l₂, l₃ and n₁, n₂, n₃ are constants. We substitute $V_{S}^{D} \left( {\theta ,G} \right)$, $V_{B}^{D} \left( {\theta ,G} \right)$ from (B.10), as well as their derivatives, into (B.8–B.9), and collect terms corresponding to θ, G. By solving the algebraic equations, we have

$$\left\{ {\begin{array}{*{20}l} {n_{1}^{*} = \frac{{\left( {a - b\omega } \right)^{2} }}{{4b\left( {\rho + \eta } \right)}}\left( {\mu + \frac{\phi r}{\rho + \varepsilon }} \right)} \hfill \\ {n_{2}^{*} = \frac{{\phi \left( {a - b\omega } \right)^{2} }}{{4b\left( {\rho + \varepsilon } \right)}}} \hfill \\ {n_{3}^{*} = \frac{{\delta^{2} n_{1}^{*} l_{1}^{*} }}{{\rho k_{1} }} + \frac{{\tau^{2} n_{2}^{*2} }}{{2\rho k_{2} }}} \hfill \\ \end{array} } \right.,$$

(B.11)

$$\left\{ \begin{aligned} l_{1}^{*} = \frac{{\omega \left( {a - b\omega } \right)}}{{2\left( {\rho + \eta } \right)}}\left( {\mu + \frac{\phi r}{\rho + \varepsilon }} \right) \hfill \\ l_{2}^{*} = \frac{{\phi \omega \left( {a - b\omega } \right)}}{{2\left( {\rho + \varepsilon } \right)}} \hfill \\ l_{3}^{*} = \frac{{\delta^{2} l_{1}^{*2} }}{{2\rho k_{1} }} + \frac{{\tau^{2} n_{2}^{*} l_{2}^{*} }}{{\rho k_{2} }} \hfill \\ \end{aligned} \right.,$$

(B.12)

Substituting (B.11) and (B.12) into (B.5) and (B.7), we get the expressions of the optimal selling price, advertising and design innovation strategies shown in Proposition 2.

The proof process of fashion level and goodwill is similar to that of Proposition 1, and are omitted here.This completes the proof. □

Proof of Proposition 3

Let $G_{\infty }^{D} = \xi_{1} \omega \left( {a - b\omega } \right) + \xi_{2} \left( {a - b\omega } \right)^{2}$ denote the steady state value of goodwill at time $t \to \infty$, differentiating $G_{\infty }^{D}$ with respect to ω yields $\partial G_{\infty }^{N} /\partial \omega = a\left( {\xi_{1} - 2b\xi_{2} } \right) - 2b\left( {\xi_{1} - b\xi_{2} } \right)\omega$. From $\partial G_{\infty }^{N} /\partial \omega = 0$, we have $\hat{\omega } = a\left( {\xi_{1} - 2b\xi_{2} } \right)/\left( {2b\left( {\xi_{1} - b\xi_{2} } \right)} \right)$. We discuss the following three scenarios,

• Scenario a $\xi_{1} \ge 2b\xi_{2}$. If $\omega \in \left( {0,\hat{\omega }} \right)$, then $\partial G_{\infty }^{N} /\partial \omega \ge 0$; If $\omega \in \left( {\hat{\omega },a/b} \right)$, then $\partial G_{\infty }^{N} /\partial \omega \le 0$.

• Scenario b $\xi_{1} \le b\xi_{2}$. If $\omega \in \left( {0,\hat{\omega }} \right)$, then $\partial G_{\infty }^{N} /\partial \omega \le 0$; If $\omega \in \left( {\hat{\omega },a/b} \right)$, then $\partial G_{\infty }^{N} /\partial \omega \ge 0$.

• Scenario c $b\xi_{2} \le \xi_{1} \le 2b\xi_{2}$. If $\omega \in \left( {0,a/b} \right)$, then $\partial G_{\infty }^{N} /\partial \omega \le 0$.

This completes the proof. □

Proof of Proposition 4

(i) Differentiating the supplier’s profit function $J_{S}^{D}$ with respect to $\omega$ yields,$\partial J_{S}^{D} /\partial \omega = 2b\varOmega_{1} \left( {a - 2b\omega } \right) + 4b\varOmega_{2} \left( {a - b\omega } \right)^{2} \left( {a - 4b\omega } \right) + 2b\varOmega_{3} \omega \left( {a - b\omega } \right)\left( {a - 2b\omega } \right)$. It is easy to prove that $\partial J_{S}^{D} /\partial \omega > 0$ for $\omega \in \left( {0,a/\left( {4b} \right)} \right)$ and $\partial J_{S}^{D} /\partial \omega < 0$ for $\omega \in \left( {a/\left( {2b} \right),a/b} \right)$. In other words, the supplier’s profit function $J_{S}^{D}$ is strictly increasing in $\omega$ on the interval $\left( {0,a/\left( {4b} \right)} \right)$, and strictly decreasing on the interval $\left( {a/\left( {2b} \right),a/b} \right)$. By the intermediate value theorem, there exists $\omega_{S} \in \left( {a/\left( {4b} \right),a/\left( {2b} \right)} \right)$ satisfying $\partial J_{S}^{D} /\partial \omega = 0$. Next, we prove the uniqueness of $\omega_{S}$. Taking the second-order derivative of $J_{S}^{D}$ with respect to $\omega$ yields

$$\partial^{2} J_{S}^{D} /\partial \omega^{2} = - 4b^{2} \varOmega_{1} - 24b^{2} \varOmega_{2} \left( {a - b\omega } \right)\left( {a - 2b\omega } \right) + 2b\varOmega_{3} \left[ {\left( {a - b\omega } \right)\left( {a - 4b\omega } \right) - b\omega \left( {a - 2b\omega } \right)} \right],$$

which follows that $\partial^{2} J_{S}^{D} /\partial \omega^{2} < 0$ for any $\omega_{S} \in \left( {a/\left( {4b} \right),a/\left( {2b} \right)} \right)$. This indicates that supplier’s profit function $J_{S}^{D}$ is strictly concave in $\omega$ on the interval $\left( {0,a/b} \right)$ and $\omega_{S}$ is the unique maximum value point.

(ii) Differentiating the fashion brand’s profit function $J_{B}^{D}$ with respect to ω yields,$\partial J_{B}^{D} /\partial \omega = - 2b\varOmega_{1} \left( {a - b\omega } \right) - 4b\varOmega_{2} \left( {a - b\omega } \right)^{3} + \varOmega_{3} \left( {a - b\omega } \right)^{2} \left( {a - 4b\omega } \right)$. It is easy to prove that $\partial J_{B}^{D} /\partial \omega < 0$ for $\omega \in \left( {a/\left( {4b} \right),a/b} \right)$. In other words, the fashion brand’s profit function $J_{B}^{D}$ is strictly decreasing on the interval $\omega \in \left( {a/\left( {4b} \right),a/b} \right)$. Since $a/\left( {4b} \right) < \omega_{S}$, it can be easily shown that the profit of the fashion brand strictly decreasing on the interval $\omega \in \left( {\omega_{S} ,a/b} \right)$.This completes the proof. □

Proof of Proposition 5

Differentiating the profit of the fashion supply chain $J_{SC}^{D}$ with respect to ω yields, $\partial J_{SC}^{D} /\partial \omega = \partial J_{B}^{D} /\partial \omega + \partial J_{S}^{D} /\partial \omega = - 2b^{2} \varOmega_{1} \omega - 12b^{2} \varOmega_{2} \omega \left( {a - b\omega } \right)^{2} + a\varOmega_{3} \left( {a - b\omega } \right)\left( {a - 3b\omega } \right)$. It is easy to prove that $\partial J_{SC}^{D} /\partial \omega < 0$ for $\omega \in \left( {a/\left( {3b} \right),a/b} \right)$. In addition, the limit value is $\mathop {\lim }\limits_{\omega \to 0} \partial J_{SC}^{D} /\partial \omega = a^{3} \varOmega_{3} > 0$. By the intermediate value theorem, there exists $\omega_{SC} \in \left( {0,a/\left( {3b} \right)} \right)$ satisfying $\partial J_{SC}^{D} /\partial \omega = 0$. Next, we prove the uniqueness of $\omega_{SC}$.Taking the second-order derivative of $J_{SC}^{D}$ with respect to ω yields $\partial^{2} J_{SC}^{D} /\partial \omega^{2} = - 2b^{2} \varOmega_{1} - 12b^{2} \varOmega_{2} \left( {a - b\omega } \right)\left( {a - 3b\omega } \right) - 2ab\varOmega_{3} \left( {2a - 3b\omega } \right)$, which follows that $\partial^{2} J_{SC}^{D} /\partial \omega^{2} < 0$ for any $\omega_{SC} \in \left( {0,a/\left( {3b} \right)} \right)$. This indicates that the profit of fashion supply chain $J_{SC}^{D}$ is strictly concave in ω on the interval $\left( {0,a/\left( {3b} \right)} \right)$ and $\partial J_{SC}^{D} /\partial \omega$ is strictly decreasing in ω. Therefore $\omega_{SC}$ is the unique maximum value point.This completes the proof. □

Proof of Proposition 6

Straightforward comparisons, using the equilibrium selling price, advertising effort and design innovation from Propositions 1 and 2, lead to the results:

$$\Delta p^{*} = p^{C*} - p^{D*} ,$$

(B.13)

$$\Delta A^{*} = A^{C*} - A^{D*} ,$$

(B.14)

$$\Delta I^{*} = I^{C*} - I^{D*} ,$$

(B.15)

$$\Delta J_{SC}^{*} = J_{SC}^{C*} - J_{SC}^{D*} .$$

(B.16)

• (i) Based on the results in Propositions 1 and 2, we can verify that $\Delta p^{*} = - \omega /2 < 0$, $\Delta A^{*} = \frac{\tau \phi }{{4bk_{2} \left( {\rho + \varepsilon } \right)}}\left[ {a^{2} - \left( {a - b\omega } \right)^{2} } \right] > 0$.

• (ii) We can also obtain $\Delta I^{*} = \frac{\delta }{{2k_{1} \left( {\rho + \eta } \right)}}\left( {\mu + \frac{\phi r}{\rho + \varepsilon }} \right)\left\{ {\frac{{a^{2} }}{2b} - \omega \left( {a - b\omega } \right)} \right\}$. For notational convenience, let $f_{1} = \omega \left( {a - b\omega } \right)$. It follows that $\partial f_{1} /\partial \omega = a - 2b\omega > 0$ for any $\omega \in \left( {0,a/\left( {3b} \right)} \right)$ and This implies that f₁ is strictly increasing on the internal $\left( {0,a/\left( {3b} \right)} \right)$. Since the maximum value of f₁ is $\mathop {\lim }\limits_{{\omega \to a/\left( {3b} \right)}} f_{1} = 2a^{2} /9b$, we get $a^{2} /2b - f_{1} > 0$. Subsequently, we can obtain $\Delta I^{*} > 0$.

• (iii) We can obtain $\Delta J_{SC}^{*} = b^{2} \omega^{2} \varOmega_{1} + \left[ {a^{4} - \left( {a - b\omega } \right)^{3} \left( {a + 3b\omega } \right)} \right]\varOmega_{2} + \left[ {a^{4} /b - a\omega \left( {a - b\omega } \right)^{2} } \right]\varOmega_{3}$. For notational convenience, let $f_{2} = \left( {a - b\omega } \right)^{3} \left( {a + 3b\omega } \right)$, $f_{3} = a\omega \left( {a - b\omega } \right)^{2}$. It follows that $\partial f_{2} /\partial \omega = - 12\omega b^{2} \left( {a - b\omega } \right)^{2} < 0$ for any $\omega \in \left( {0,a/\left( {3b} \right)} \right)$. This implies that f₂ is strictly decreasing on the internal $\left( {0,a/\left( {3b} \right)} \right)$. Since the maximum value of f₂ is $\mathop {\lim }\limits_{\omega \to 0} f_{2} = a^{4}$, we get $a^{4} - f_{2} > 0$. Similarly, we get $df_{3} /d\omega = a\left( {a - b\omega } \right)\left( {a - 3b\omega } \right) > 0$ for any $\omega \in \left( {0,a/\left( {3b} \right)} \right)$. This implies that f₃ is strictly increasing on the internal $\left( {0,a/\left( {3b} \right)} \right)$. Since the maximum value of f₃ is $\mathop {\lim }\limits_{{\omega \to a/\left( {3b} \right)}} f_{3} = 4a^{4} /27b$, we get $a^{4} /b - f_{3} > 0$. Subsequently, we can obtain $\Delta J_{SC}^{*} > 0$.

This completes the proof. □

Proof of Proposition 7

Let $V_{S}^{U} \left( {\theta ,G} \right)$, $V_{B}^{U} \left( {\theta ,G} \right)$ denote the value function for the supplier and the fashion brand, respectively. The supplier’s HJB equation can be specified as

$$\rho V_{S}^{U} \left( {\theta ,G} \right) = \mathop {\hbox{max} }\limits_{I} \left\{ {\varPsi - \frac{1}{2}k_{1} I^{2} + V_{S\theta }^{U'} \left( {\delta I - \eta \theta } \right) + V_{SG}^{U'} \left( {\tau A + r\theta - \varepsilon G} \right)} \right\},$$

(B.17)

The equilibrium design innovation is given below by maximizing the right-hand side of (B.17) with respect to I, i.e.,

$$I = \frac{{\delta V_{SC\theta }^{U'} }}{{k_{1} }}.$$

(B.18)

The fashion brand’s HJB equation can be specified as

$$\rho V_{B}^{U} \left( {\theta ,G} \right) = \mathop {\hbox{max} }\limits_{p,A} \left\{ {p\left( {a - bp} \right)\left( {\mu \theta + \phi G} \right) - \varPsi - \frac{1}{2}k_{2} A^{2} + V_{B\theta }^{U'} \left( {\delta I - \eta \theta } \right) + V_{BG}^{U'} \left( {\tau A + r\theta - \varepsilon G} \right)} \right\}$$

(B.19)

and its maximisation provides the necessary condition for selling price and advertising efforts:

$$p = \frac{a}{2b},A = \frac{{\tau V_{SCG}^{U'} }}{{k_{2} }}.$$

(B.20)

Similar to that of Proposition 2, we have

$$\left\{ {\begin{array}{*{20}l} {I^{U*} = \frac{{\delta a^{2} }}{{4bk_{1} \left( {\rho + \eta } \right)}}\left( {\mu + \frac{\phi r}{\rho + \varepsilon }} \right)} \hfill \\ {A^{U*} = \frac{{\tau \phi a^{2} }}{{4bk_{2} \left( {\rho + \varepsilon } \right)}}} \hfill \\ {p^{U*} = \frac{a}{2b}} \hfill \\ \end{array} } \right..$$

(B.21)

Consequently, $I^{U*} = I^{C*}$, $p^{U*} = p^{C*}$, $A^{U*} = A^{C*}$. These equilibrium strategies are identical with integrated solutions. Similarly, the profits of the supplier and fashion brand, $J_{B}^{U*} \left( \varPsi \right)$, $J_{S}^{U*} \left( \varPsi \right)$, are given by the value functions in (27) and (28).This completes the proof. □