Horizontally Differentiated Store Brands: Production Outsourcing to National Brand Manufacturers

Table 5

Equilibrium outcome under ARM scenario with assortment 𝐹.

	For 𝛽<𝛽	For 𝛽≥𝛽
𝑝𝑛	−2√3𝛽𝛽^4−4𝛽^2+3−5𝛽^3−5𝛽^2+15𝛽+158𝛽+13−𝛽^2	13𝛽+716(𝛽+1)
𝑝𝑠	−2√3𝛽^4−4𝛽^2+3+𝛽^4−5𝛽^3−9𝛽^2+15𝛽+188𝛽+13−𝛽^2	−2𝛽^3+10𝛽^2+15𝛽−316𝛽𝛽+1
𝑤𝑛	−2√3𝛽𝛽^4−4𝛽^2+3−3𝛽^3−3𝛽^2+9𝛽+96𝛽+13−𝛽^2	3𝛽+14𝛽+1
𝑤𝑠	−√3𝛽^4−4𝛽^2+3+3𝛽+36𝛽+6	−𝛽^3+5𝛽^2+7𝛽−38𝛽^2+8𝛽
𝑚𝑛	2√3𝛽𝛽^4−4𝛽^2+3−3𝛽^3−3𝛽^2+9𝛽+924𝛽+13−𝛽^2	𝛽+316𝛽+1
𝑚𝑠	3𝛽^4−3𝛽^3−1−𝛽𝛽+13−𝛽^24√3𝛽^2−6√3−15𝛽^2+9𝛽+1824𝛽+13−𝛽^2	𝛽+316𝛽𝛽+1
𝑞𝑠	√3𝛽^4−4𝛽^2+3−𝛽^2+34𝛽+13−𝛽^2	𝛽+316𝛽(𝛽+1)
𝑞1	−4√3𝛽𝛽^4−4𝛽^2+3+3𝛽^3−3𝛽^2−9𝛽+924𝛽+13−𝛽^2	0
𝑞2 (=𝑞3)	2√3𝛽𝛽^4−4𝛽^2+3−3𝛽^3−3𝛽^2+9𝛽+924𝛽+13−𝛽^2	𝛽+316(𝛽+1)
𝜋1	4√3(1−𝛽)(1+𝛽)(3−𝛽^2)(6−5𝛽)+41𝛽^3−47𝛽^2−63𝛽+81192(𝛽+1)(3−𝛽^2)	𝛽+3^2256𝛽^2𝛽+1^2
𝜋𝑀	316	𝛽+311𝛽^3+9𝛽^2+7𝛽−3128𝛽^2𝛽+1^2

Where 𝛽≈0.448 is the unique feasible solution for 13𝛽³+19𝛽²+9𝛽−9=0.