Budget Constrained Cournot Duopoly
I am currently working on this problem as an extension to a paper I am working on. The premise revolves around analyzing European football clubs as win maximizing as opposed to profit maximizing firms. Since clubs have no control over whether they win games or not, instead under the model clubs seek to maximize the number of player talents on their team (x_i) subject to some nonzero profit constraint (pi_i). We model revenue (R_i) as a function of market value/size of the club (m_i) and win percentage of the club (w_i). We specify win percentage as a function increasing in (x_i), decreasing in the number of player talents of the other teams (x_j), and increasing in the wage salary level of the club_i (c_i). Since for each club the wagesalary level may be different from club to club, we model costs simply as =(c_i)(x_i). And so our model is as follows
MAX[x_i] SUBJ [R_i(m_i,w_i(x_i,x_j,c_i))(c_i)(x_i)=pi_i]
In addition we will assume that each club_i will take as given x_j, therefore giving us our template of a Cournot equilibrium. Moreover, we will limit ourselves to n=2.
I know that we have to set up this expression as a Langrangian; however, my question is as to whether to treat lambda as a choice variable. It makes sense to do so since we are concerned only with admissible x_i's as allowed by our profit constraint. We know our profit is binding since otherwise each club would seek to buy an ad infinitum player talents. And so we know by Weierstrass a set of optima exists, and if we impose concavity on our revenue function (as the paper does), we get some unique optimum x*_i. My question revolves around second order conditions. Specifically, am I constructing a bordered Hessian for this problem? It doesn't make sense in my mind since I am just optimizing for club_i. But then how do I check to make sure my optima are admissible by our budget constraint? Is that not the purpose of the bordered Hessian? I think I am overcomplicating the problem, but at the same time, I want to make sure I get this right. Even if I treat this as a single choice variable optimization with some budget constraint and show through casebycase analysis that our profit constraint is binding, then what the fuck is (are) my second order condition(s)?
