Cooperative Advertising in a Supply Chain with Horizontal Competition

Table 1

Notation.

𝑡	Time 𝑡, 𝑡≥0
𝐺𝑖(𝑡)	Goodwill of the product 𝑖 at time 𝑡, 𝑖∈{1,2}
𝑈𝑀𝑖(𝑡)	Manufacturer 𝑖’s national advertising efforts at time 𝑡
𝑈𝑅𝑖(𝑡)	Retailer’s local advertising efforts for product 𝑖 at time 𝑡
𝑆𝑖(𝑡)	Sales of product 𝑖 along time 𝑡
𝜃∈[0,1]	The national advertising’s competition coefficient
𝜒∈[0,1]	The local advertising’s competition coefficient
𝜙𝑖∈[0,1]	Manufacturer 𝑖’s participation rate for the retailer’s advertising cost
𝜌𝑀𝑖≥0	Marginal profit of manufacturer 𝑖
𝜌𝑅𝑖≥0	Marginal profit of the retailer sells the product 𝑖
𝛿>0	Diminishing rate of goodwill
𝑟>0	Discount rate of the manufacturers and the retailer
𝜋𝑀𝑖, 𝜋𝑅	Profit functions for 𝑀𝑖 and 𝑅, respectively
𝐽𝑀𝑖, 𝐽𝑅	Current value of profit functions for 𝑀𝑖 and 𝑅, respectively