Equilibrium search with heterogeneous firms, workers and endogenous human capital

This article shows how the endogenous human capital affects the labor market equilibrium when jobs provided by firms can be either unskilled or skilled and workers differ in their education level which can be either low-educated or high-educated. We develop an equilibrium search model in which the high-educated workers are assumed to be able to accept either the unskilled jobs or the skilled jobs, while the low-educated workers can only accept the unskilled jobs. The market equilibrium is characterized by deriving the unemployment rate and the human capital distributions when the growth rate of the human capital is an endogenous variable. The results demonstrate that the structure proportion of the offered jobs affects the equilibrium which shows there is a threshold that can distinguish whether the equilibrium is separating or cross-skill. In addition, the cross-skill equilibrium solution implies the high-educated workers are more likely to own higher pay rates than the low-educated workers with same tenure. It also yields a new insight on the effect of the structure proportion of workers on the profits, which implies the profits of the firms decrease with the increasing number of the low-educated workers. Moreover, the profit of the firms offering the skilled jobs is greater than those offering the unskilled jobs until there is only very few high-educated workers.