Horizontally Differentiated Store Brands: Production Outsourcing to National Brand Manufacturers

Table 9

Equilibrium outcome under MAR scenario.

	For 0<𝛽<𝛽1	For 𝛽1<𝛽<𝛽	For 𝛽<𝛽<1
𝑝𝑛	58	6+6𝛽−2𝛽^2−2𝛽^3−3𝛽(6+4𝛽−4𝛽^2−3𝛽^3+𝛽^5)2(1+𝛽)(3−𝛽^2)	12−2𝛽^2−3𝛽63−𝛽^2
𝑝𝑠	𝛽+48	3+2𝛽^2−2𝛽^3−𝛽^4+3𝛽2−𝛽6+4𝛽−4𝛽^2−3𝛽^3+𝛽^52(1+𝛽)(3−𝛽^2)	9−2𝛽^363−𝛽^2
𝑤𝑛	12	1+𝛽3−𝛽^2−2𝛽6+4𝛽−4𝛽^2−3𝛽^3+𝛽^5(1+𝛽)(3−𝛽^2)	12
𝑤𝑠	𝛽2	𝛽1+𝛽3−𝛽^2−2𝛽6+4𝛽−4𝛽^2−3𝛽^3+𝛽^5(1+𝛽)(3−𝛽^2)	𝛽3
𝑚𝑛	12	𝛽(6+4𝛽−4𝛽^2−3𝛽^3+𝛽^5)2(1+𝛽)(3−𝛽^2)	𝛽^2−3𝛽+363−𝛽^2
𝑚𝑠	4−3𝛽8	3−4𝛽^2+𝛽^4+𝛽𝛽(6+4𝛽−4𝛽^2−3𝛽^3+𝛽^5)2(1+𝛽)(3−𝛽^2)	3−2𝛽23−𝛽^2
𝑞𝑠	12(𝛽+1)	12(1+𝛽)	3−2𝛽23−𝛽^2
𝑞1	1−3𝛽8(𝛽+1)	𝛽^3−3𝛽+𝛽(6+4𝛽−4𝛽^2−3𝛽^3+𝛽^5)2(1+𝛽)(3−𝛽^2)	—
𝑞2 (=𝑞3)	18	𝛽(6+4𝛽−4𝛽^2−3𝛽^3+𝛽^5)2(1+𝛽)(3−𝛽^2)	𝛽^2−3𝛽+363−𝛽^2
𝜋1	17−15𝛽64(𝛽+1)	3−2𝛽^243−𝛽^22	3−2𝛽^243−𝛽^22
𝜋𝑀	316	−36𝛽+12𝛽^2+12𝛽^3+6𝛽^4−6𝛽^5+3(3−𝛽^2)𝛽(6+4𝛽−4𝛽^2−3𝛽^3+𝛽^5)2(1+𝛽)(3−𝛽^2)^2	16

Where 𝛽₁≈0.107 is the unique feasible solution for 𝜋^AMR(F)₁=𝜋^AMR(S)₁ in 𝛽<1/7 (equivalent to 15𝛽⁵−17𝛽⁴−26𝛽³−26𝛽²+87𝛽−9=0) and 𝛽≈0.215 is the unique feasible solution for 𝜋^MAR_𝑀^(𝐹)(𝛽)=𝜋^MAR(S)_𝑀(𝛽) (equivalent to 9+117𝛽−42𝛽³−17𝛽⁴+19𝛽⁵−27𝛽(6+4𝛽−4𝛽²−3𝛽³+𝛽⁵)+3𝛽²(−14+3𝛽(6+4𝛽−4𝛽²−3𝛽³+𝛽⁵))=0), respectively.