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Coordinating pricing and advertising in a two-period fashion supply chain

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Abstract

Firms in the fashion industry frequently launch new styles of their products, thereby leading to a ‘two-period’ phenomenon in product sales: the normal selling time and the last salvage time. Researches about cooperative (co-op) advertising, however, seldom concentrate on this phenomenon. This study explores co-op advertising in a two-period fashion supply chain system in which the retailer offers a price discount in the second period. We derive the optimal pricing, product design level and advertising efforts in two scenarios: a decentralised scenario with a co-op advertising program and integrated scenario. Additionally, we propose a bilateral participation-revenue-sharing contract to coordinate this channel. Some insights and management implications are obtained. Firstly, if the consumers are insensitive to the first-period price, the retailer will set a high retail price in this period, while selling the product at a big discount in the second period to stimulate extra consumption. Secondly, if the price elasticity of demand is small, the retailer will spend more on the advertisement, otherwise the condition is opposite. Lastly, this study verifies that the bilateral participation-revenue-sharing contract can achieve a seamless coordination in this supply chain with a transfer payment.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Numbers 71502076, 71872075).

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Correspondence to Yi He.

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Appendix

Appendix

Proof of the equilibrium solutions in the decentralised scenario

The decision sequences of the supply chain members are: first, the manufacturer determines the product design level and subsidy rate \( \phi_{i} (i = 1,2) \) at the beginning of the first period; second, the retailer decides advertising efforts a1 and retail price p in the first period; third, the retailer decides advertising efforts a2 and price deduction percentage θ at the beginning of the second period.

According to the backward induction, we first solve the price deduction percentage and the advertising efforts in the second period for the retailer; the first-order conditions are as follows:

$$ \left\{ {\begin{array}{*{20}l} {\partial \pi _{r}^{D} /\partial \theta = - 2(\gamma \sqrt {a_{1} } + \sqrt {a_{2} } )\lambda (1/2 + ((\theta - 1)p + w_{2} /2)\beta _{2} )p} \hfill \\ {\partial \pi _{r}^{D} /\partial a_{2} = (\beta _{2}^{2} \lambda w^{2} + 8\phi _{2} \beta _{2} \sqrt {a_{2} } - 2\beta _{2} \lambda w - 8\beta _{2} \sqrt {a_{2} } + \lambda )/8\beta _{2} \sqrt {a_{2} } .} \hfill \\ \end{array} } \right. $$
(A.1)

The second-order conditions are as follows:

$$ \left\{ {\begin{array}{*{20}l} {\partial ^{2} \pi _{r}^{D} /\partial \theta ^{2} = - 2\lambda \beta _{2} p^{2} (\sqrt {a_{2} } + \gamma \sqrt {a_{1} } ) < 0} \hfill \\ {\partial ^{2} \pi _{r}^{D} /\partial \theta \partial a_{2} = - \lambda p(1/2 + ((\theta - 1)p + w/2)\beta _{2} )/\sqrt {a_{2} } } \hfill \\ {\partial ^{2} \pi _{r}^{D} /\partial a_{2}^{2} = - \lambda (1 - \beta _{2} w)^{2} /16\beta _{2} a_{2}^{{3/2}} < 0} \hfill \\ {\partial ^{2} \pi _{r}^{D} /\partial a_{2} \partial \theta = 0.} \hfill \\ \end{array} } \right. $$
(A.2)

We then can obtain the Hessian Matrix

$$ H = \left[ \begin{aligned} \frac{{\partial^{2} \pi_{r}^{D} }}{{\partial \theta^{2} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\partial^{2} \pi_{r}^{D} }}{{\partial \theta \partial a_{2} }}{\kern 1pt} {\kern 1pt} \hfill \\ \frac{{\partial^{2} \pi_{r}^{D} }}{{\partial a_{2}^{2} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\partial^{2} \pi_{r}^{D} }}{{\partial a_{2} \partial \theta }}{\kern 1pt} \hfill \\ \end{aligned} \right], $$

and Δ1 = <0, Δ2 > 0. Therefore, H is negative definite. Therefore, the profit function has a maximum.

Then, we can obtain the expressions of θ and a2 by letting the first-order conditions to be zero. The price deduction percentage and the advertising efforts are as follows:

$$ \theta^{D} = 1 - \frac{{\beta_{2} w + 1}}{{2\beta_{2} p^{D} }}, $$
(A.3)
$$ a_{2}^{D} = \frac{{\lambda^{2} (1 - \beta_{2} w)^{4} }}{{64\beta_{2}^{2} (1 - \phi_{2} )^{2} }}. $$
(A.4)

Then, we solve the optimal advertising efforts and retail price in the first period, a1 and p. Substituting Eqs. (A.3) and (A.4) into Eq. (9), we can obtain the retailer’s profit function \( \pi_{r}^{D} (a_{1} ,p) \). To maximize the profit, we can obtain the first-order conditions of a1 and p

$$ \frac{{\partial \pi_{r}^{D} (a_{1} ,p)}}{{\partial a_{1} }} = \frac{{\lambda (1 - \beta_{1} w)^{2} }}{{8\beta_{1} \sqrt {a_{1} } }} + \frac{{\lambda \gamma (1 - \beta_{2} w)^{2} }}{{8\beta_{2} \sqrt {a_{1} } }} + \phi_{1} - 1, $$
(A.5)

and

$$ \frac{{\partial \pi_{r}^{D} (a_{1} ,p)}}{\partial p} = - 2\lambda \beta_{1} p\sqrt {a_{1} } { + }\lambda \beta_{1} w\sqrt {a_{1} } - 2\beta_{1} p\delta \sqrt x + \beta_{1} w\delta \sqrt x { + }\lambda \sqrt {a_{1} } + \delta \sqrt x . $$
(A.6)

The second-order conditions are:

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\partial ^{2} \pi _{r}^{D} (a_{1} ,p)}}{{\partial a_{1}^{2} }} = - \frac{{4\lambda \beta _{2} (1 - \beta _{1} p)(p - w)}}{{16\beta _{2} a_{1}^{{3/2}} }} < 0} \hfill \\ {\frac{{\partial ^{2} \pi _{r}^{D} (a_{1} ,p)}}{{\partial a_{1} \partial p}} = 0} \hfill \\ {\frac{{\partial ^{2} \pi _{r}^{D} (a_{1} ,p)}}{{\partial p^{2} }} = - 2\beta _{1} (\delta \sqrt x + \lambda \sqrt {a_{1} } ) < 0} \hfill \\ {\frac{{\partial ^{2} \pi _{r}^{D} (a_{1} ,p)}}{{\partial p\partial a_{1} }} = - \frac{{( - 1/2 + (p - w/2)\beta _{1} )\lambda }}{{\sqrt {a_{1} } }}} \hfill \\ \end{array} } \right. $$
(A.7)

We then can obtain the Hessian Matrix

$$ H = \left[ \begin{aligned} \frac{{\partial^{2} \pi_{r}^{D} (a_{1} ,p)}}{{\partial a_{1}^{2} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\partial^{2} \pi_{r}^{D} (a_{1} ,p)}}{{\partial a_{1} \theta p}}{\kern 1pt} {\kern 1pt} \hfill \\ \frac{{\partial^{2} \pi_{r}^{D} (a_{1} ,p)}}{{\partial p^{2} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\partial^{2} \pi_{r}^{D} (a_{1} ,p)}}{{\partial p\partial a_{1} }}{\kern 1pt} \hfill \\ \end{aligned} \right], $$

and Δ1 = <0, Δ2 > 0. Therefore, H is negative definite. Therefore, the profit function has a maximum.

Therefore, we can obtain the expressions of a1 and p by letting the first-order conditions to be zero. The optimal advertising efforts and retail price in the first period are

$$ a_{1}^{D} = \frac{{\lambda^{2} (\gamma \beta_{1} (1 - \beta_{2} w)^{2} + \beta_{2} (1 - \beta_{1} w)^{2} )^{2} }}{{64\beta_{1}^{2} \beta_{2}^{2} (1 - \phi_{1} )^{2} }}, $$
(A.8)
$$ p^{D} = \frac{{\beta_{1} w + 1}}{{2\beta_{1} }}. $$
(A.9)

Finally, substituting Eqs. (A.3), (A.4), (A.8) and (A.9) into Eq. (9), we can obtain the manufacturer’s profit function \( \pi_{m}^{D} (x,\phi_{1} ,\phi_{2} ) \). To maximize the profit, we can obtain the first-order conditions of x, ϕ1 and ϕ2: \( \partial \pi_{m}^{D} (x,\phi_{1} ,\phi_{2} )/\partial x,\partial \pi_{m}^{D} (x,\phi_{1} ,\phi_{2} )/\partial \phi_{1} \) and \( \partial \pi_{m}^{D} (x,\phi_{1} ,\phi_{2} )/\partial \phi_{2} \), respectively. And the Hessian Matrix is negative definite. Therefore, we can obtain the expressions of x, ϕ1 and ϕ2 by letting the first-order conditions to be zero. The optimal product design level and subsidy rates are

$$ x^{D} = \left( {\frac{{\delta (w - c)(1 - \beta_{1} w)}}{4t}} \right)^{2} , $$
(A.10)
$$ \phi_{1}^{D} = \frac{5A + B - 4C - 6D}{3A - B - 4C - 2D}, $$
(A.11)
$$ \phi_{2}^{D} { = }\frac{{5\beta_{2} w - 4\beta_{2} c - 1}}{{3\beta_{2} w - 4\beta_{2} c + 1}}, $$
(A.12)

where \( A = \beta_{1} \beta_{2}^{2} \gamma w^{2} + \beta_{1}^{2} \beta_{2} w^{2} \), \( B = \beta_{1} \gamma + \beta_{2} \), \( C = \beta_{1} \beta_{2}^{2} c\gamma w + \beta_{1}^{2} \beta_{2} cw - \beta_{1} \beta_{2} c\gamma \) and \( D = \beta_{1} \beta_{2} \gamma w + \beta_{1} \beta_{2} w \).

Proof of the equilibrium solutions in the integrated scenario

The profit function of the integrated system is

$$ \pi^{I} = \pi_{m} + \pi_{r} = (p - c)D_{1} + ((1 - \theta )p - c)D_{2} - tx - a_{1} - a_{2} . $$

Similarly, according to the backward induction, we first solve the price deduction percentage and the advertising efforts in the second period for the retailer. The first-order conditions are given by

$$ \left\{ \begin{aligned} \partial \pi_{r}^{I} /\partial \theta = - p\lambda (\gamma \sqrt {a_{1} } + \sqrt {a_{2} } )(1 + (2(\theta - 1)p + c)\beta_{2} ) \hfill \\ \partial \pi_{r}^{I} /\partial a_{2} = (\beta_{2}^{2} \lambda c^{2} - (2c\lambda + 8\sqrt {a_{2} } )\beta_{2} + \lambda )/8\beta_{2} \sqrt {a_{2} } \hfill \\ \end{aligned} \right.. $$
(A.13)

The second-order conditions are as follows:

$$ \left\{ {\begin{array}{*{20}l} {\partial ^{2} \pi _{r}^{I} /\partial \theta _{{}}^{2} = - 2\lambda \beta _{2} p^{2} (\sqrt {a_{2} } + \gamma \sqrt {a_{1} } ) < 0} \hfill \\ {\partial ^{2} \pi _{r}^{I} /\partial \theta \partial a_{2} = - \lambda p(1 + (2(\theta - 1)p + c)\beta _{2} )/2\sqrt {a_{2} } } \hfill \\ {\partial ^{2} \pi _{r}^{I} /\partial a_{2}^{2} = - \lambda (1 - \beta _{2} c)^{2} /16\beta _{2} a_{2}^{{3/2}} < 0} \hfill \\ {\partial ^{2} \pi _{r}^{I} /\partial a_{2} \partial \theta = 0.} \hfill \\ \end{array} } \right. $$
(A.14)

We then can obtain the Hessian Matrix

$$ H = \left[ \begin{aligned} \frac{{\partial^{2} \pi_{r}^{I} }}{{\partial \theta^{2} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\partial^{2} \pi_{r}^{I} }}{{\partial \theta \partial a_{2} }}{\kern 1pt} {\kern 1pt} \hfill \\ \frac{{\partial^{2} \pi_{r}^{I} }}{{\partial a_{2}^{2} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\partial^{2} \pi_{r}^{I} }}{{\partial a_{2} \partial \theta }}{\kern 1pt} \hfill \\ \end{aligned} \right], $$

and Δ1 = <0, Δ2 > 0. Therefore, H is negative definite. Therefore, the profit function in the integrated scenario has a maximum.

Then, we can obtain the expressions of θ and a2 by letting the first-order conditions to be zero. The price deduction percentage and the advertising efforts in the integrated scenario are as follows:

$$ \theta^{I} = \frac{{\beta_{2} - \beta_{1} }}{{\beta_{2} (\beta_{1} c + 1)}}. $$
(A.15)
$$ a_{2}^{I} = \frac{{\lambda^{2} (1 - \beta_{2} c)^{4} }}{{64\beta_{2}^{2} }}. $$
(A.16)

Similarly, we get the optimal advertising efforts and retail price of the retailer and product design level of the manufacturer, a1, p and x.

$$ a_{1}^{I} = \frac{{\lambda^{2} (\beta_{1} \gamma (1 - \beta_{2} c)^{2} + \beta_{2} (1 - \beta_{1} c)^{2} )^{2} }}{{64\beta_{1}^{2} \beta_{2}^{2} }}. $$
(A.17)
$$ p^{I} = \frac{{\beta_{1} c + 1}}{{2\beta_{1} }}. $$
(A.18)
$$ x^{I} = \frac{{\delta^{2} (1 - \beta_{1} c)^{4} }}{{64\beta_{1}^{2} t^{2} }}. $$
(A.19)

Similarly, we can obtain the equilibriums in the scenario with Bilateral participation-revenue-sharing contract.

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Yan, G., He, Y. Coordinating pricing and advertising in a two-period fashion supply chain. 4OR-Q J Oper Res (2019) doi:10.1007/s10288-019-00417-x

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Keywords

  • Cooperative advertising
  • Pricing
  • Coordination
  • Two-period

Mathematics Subject Classification

  • 90B60