This paper presents a general equilibrium assignment model of workers to tasks with endogenous human capital formation and multidimensionality of skills. The model has 2 key features. First, skills are endogenous and multidimensional. Second, two types of assignment occur, workers self-select their education and firms assign workers to tasks/machines. This assignment model yields two functions mapping skills of each type to tasks. Equilibrium is characterized by different wage functions for each type of skills, so that the wage distributions generally overlap. This model offers a unique framework to analyze changes in the wage structure within and between skills groups of workers and distinguishes between technological change that is related to machines (the technical factor) or related to workers (the human factor). I show both theoretically and through simulations that the model can reproduce simultaneously i) the overlap in the wage distributions of college and high-school graduates, ii) the rise in the college-premium, iii) the rise in within wage inequality iv) the differential behavior of the between and within wage inequality in the 60s and 70s and, v) the decline of the wage at the first decile of the overall wage distribution. A family of closed form solutions for the wage functions is proposed. In this family, the output of worker-task pairs is Cobb-Douglas, tasks are distributed according to a Beta distribution and the mapping functions have a logistic form.
Endogenous human capital formation tasks assignment substitution technical change and wage distribution