Debraj Ray Income Distribution and Macroeconomic Behavior

There are an infinite number of generations, one at each date. In each period t a member of generation receives an income w, which depends on the proportions of the population in different skill categories, as well as the skill category to which this member belongs. This income is partly consumed and partly used in educating the offspring of this member. Depending on the education level of the child, the child receives an income next period, and the entire process repeats itself. At each date, a production function f determines the wage to skill categories. Assume that there are only two skill categories: “high” and “low”. Being low at any date requires no investment by the parent; being high requires an investment of x, which is denominated in terms of final output and kept fixed for the purpose of the exercise. If there is a unit mass of individuals, and a fraction λ of them is high at some date, then the high wage is given by