Equilibrium in Matching Models with Employment Dependent Productivity

In a standard search and matching framework, the labor market presents frictions while in the competitive product market the demand is infinitely elastic.To have a more realistic framework, some models abandon the assumption of infinite elasticity and consider a two-tier productive scheme in the goods market. In this paper, I establish the conditions that are sufficient for the existence and the uniqueness of a steady-state equilibrium for this kind of models. I also notice that some standard assumptions about the production and matching technology (a Cobb-Douglas function) do not fulfill such conditions and so may hinder the existence of an equilibrium.