Systems of nonlinear algebraic equations with positive solutions

We are concerned with the positive solutions of an algebraic system depending on a parameter α> 0 and arising in economics. For α> 1 we prove that the system has at least a solution. For 0<α<1 we give three proofs of the existence and a proof of the uniqueness of the solution. Brouwer’s theorem and inequalities involving convex functions are essential tools in our proofs.