“…It should be noted that the feasible set in the welfare optimization problems may not be convex and therefore the first-order conditions above are necessary, but not sufficient for the solution to be a maximum (see Abio et al, 2004, Conde-Ruiz et al, 2010, Fanti and Gori, 2012, De La Croix et al, 2012, Yew and Zhang, 2013and Li and Zhang, 2015. Thus for the solution to be a social optimum we assume:…”