this is more likely to be an economy question.

Let us take the example of a benevolent parent who wishes to maximise the well-being of hem-{his child. Suppose that the welfare of the child can be written in the following way: W{S,E,C}=ckln{8)+bxln(E+E)+(lab)x1:1(CQ) where 5' (2 0) is the amount of time that the parent spends with/on the child, C (2 0) is consumption, E (2 0) is private education, As for the parameters, Q i} 0) stands for subsistence consumption level, L: {2 0] denotes publicly provided education, a. {E [0, 1]] and b [E [0, 1]} are weights. ((1 + b E [i], 1]) The parent has to allocate T units of time between the child and income-earning activity such that herfhis time constraint is: S + L g: T (TO) where L (2 [l] is the amount of time spent on an income-earning activity. Furthermore, the parent faces the following budget constraint: ch + was 5 wt {30) where pg (> 0) is the unit price of consampticn, pg (> 0) is the unit price of education, and w (> 0) is income per unit of time spent on the income-eaming activity. Finally, the amount of private education cannot be negative: E 2 U, 1. Why are the time constraint and the budget constraint both binding at the optimum? Substitute the two constraints into the objective function such that the child's welfare is a function of time spent with/on the child S and private education E. 2. Under what condition is it optimal to provide no private education (5" = D}? What is the optimal amount of time spent withfon the child 5" in this case (as a function of the parameters}? Comment. 3. Under what condition is it optimal to provide some private education {E' > 0)? What is the optimal amount of private education E" and the optimal amount of time spent with/Won the child S\" in this case (as a function of the parameters)? Conuneut