I think this is a differential question, but I have no idea where to start if it is.
Peoples productivity (v) is a function of two attributes - innate ability (a) and the level of education (e)
v:=v(a.e)
Innate ability is given by genetic factors - it does not cost anything. Education, however, has a cost which depends on the level of education but also on innate ability.
Thus the cost of education is c:= c(a,e)
It is plausible to assume that more education costs more, but the greater the ability the lower the cost. Thus the partial derivative of the cost function are positive and negative, respectively
Ce > 0 and Ca< 0
In addition we assume that the cross-partial derivative cea<0
meaning that the marginal cost of education is lower for those with higher ability.
Since ability is not obserable, but the level of education is, employers' pay higher wages to those with a higher level of education. If the wage schedule is w(e), persons chose the level of education to maximise
W(e) – c(a,e)
a) Show that the more able would undertake more education, i.e. show that de/da>0 where e refers to the optimal value of education.
In equilibrium the wages employers' pay must be commensurate with the productivity of workers. For this to be the case there must be a function relating ability to the level of eduction. Denoting this function by a=a(e), the equilibrium condition is
W(e) = v(a(e),e)
(b) Show that this implies that persons would 'overinvest', in education in the sense that at the optimal value of e the marginal cost of education ce would
exceed the marginal benefit ve
Help will be greatly appreciated