Lets define a variable xt capturing the households relative human capital endowment with respect to the economys average, such that

Show that relative human capital evolves according to

In period t, a parental household equipped with human capital ht earns a labour income of wht, where w>0 represents the constant wage rate. This household derives utility out of own consumption (ct), the number of children nt and their level of human capital ht+1. Education is provided by teachers who are equipped with the economy's average level of human capital ht. Human capital per child evolves from one period to another according to ht+1=(et+e)ht,00 is a constant parameter and et represents the level of education per child. The households' utility function is specified as Ut=lnct+ln(ntht+1) with >0 Raising one child to adulthood requires a share of 0