Two firms, unimaginatively named 1 and 2 , can put on advertising campaigns with effectiveness eᵢ at a price of 1/2ke²ᵢ, κ>0. Here, we are thinking of "effectiveness" as a portion of the potential buyers that are reached and form a positive image of the company's good. Associated with their advertising effectivenesses are the extra profits, which are given by
π₁(e₁, e₂) = v(e₁ - e₂) - 1/2ke²₁, and
π₂(e₁, e₂) = v(e₂ - e₁) -1/2ke²₂
where v(0) = 0, v'(r) > 0, and v"(r) < 0.
1. Show that there is a unique equilibrium, and that both fims hate it.