A Mathematical Model for the Origin of Name Brands and Generics

The candidate solution reproduces the paper’s piecewise-linear advertising model dai/dt = B(ai|a) − ka with µ = ν = 1, using the same expression for B(ai|a) derived from the profit-maximizing price/quantity and the piecewise Qfree(ai|a) definition. It derives: (i) the existence condition for a differentiated (bimodal) state via maxai ∂πi/∂ai > 0, matching Eq. (2.13) (large N) and correctly noting the (N−1)/N factor from Eq. (2.12) for finite N; (ii) the self-consistency a = λ(1 − x)/x and the nonnegativity of the unstable fixed point to obtain the same xcrit formula as Eq. (2.16); (iii) the undifferentiated-state stability condition ∂πi/∂ai|ai=a < 0 (Eq. (2.21)); and (iv) the bistability window B(a|a) < ka < maxa B(a|a) (Eq. (2.22)), equivalently ∂πi/∂ai|ai=a < 0 < maxa ∂πi/∂ai (Eq. (2.23)). All these steps, expressions, and conclusions align with the paper’s derivations and logic, including where the maximum of B is attained at the right-hand boundary of the linear region and the role of the unstable interior zero. Minor differences are purely presentational; the arguments are essentially the same and rely on the same one-dimensional flow and sign considerations. See the paper’s Eqns. (2.6), (2.11)–(2.16), (2.19)–(2.23) for the corresponding statements and proofs .