$(10$pts $)$Consider the three$-$period model of early child education with a single school district containing two schools, A and B$,$where F$1$A$>$F$1$B$($and$,$hence$,$ Q$1$A$>$Q$1$B$).$The school district's maximization problem is

maxS$2$A$,$S$2$BU$=$Q$2$A$+$Q$2$B

subject to the budget constraint

S$2$A$+$S$2$B$=$ar$($S$)2$

Explain under what conditions $($mathematically as well as intuitively$)$on the period $2$EPF $,$the utility$-$maximizing strategy of the school district is

Equality: S$2$A$=$S$2$B

Compensation: S$2$A$>$S$2$BS$2$A

Reinforcement: S$2$A$>$S$2$B