Consider a parent who has preferences given by:Vp=wp(1l) +zVc(Bc),where wp is the wage of the parent, l is the time spent on increasing the patience ofthechild, z isthealtruismweightandsatisfies 0< z <1,and Vc(Bc) is the utility of the child as a function of the child's patience BC.the "production function" for the child's patience is given by: Bc=l^

where0< <1. The child's utility is given by:Vc(Bc) =w^Yi+Bcw^Oi, where w^Yi is the wage when young, w^Oi is the wage when old, and i denotes the occupation chosen by the child.There are two possible occupations:An actor has an income profile of w^Yactor= 2 and w^Oactor= 2, whereas a dentist has a profile of w^Ydentist= 0 and w^Odentist= 6.The career is chosen by the child given his or her patience Bc.

(a) Solve the occupational choice problem of the child. That is, find a level B such that the child chooses to be a dentist if Bc > B.

(b) Solve the parent's problem of investing in the child's patience conditional on the child's occupation.That is, find the optimal Bc for a parent who knows for sure that the child will be an actor, and similarly for a parent who knows that the child will a dentist for sure. Explain how and why the solutions are different.

(c) Now consider the setting with occupational choice (i.e., the child will choose its occupation based on its Bc).Let the parameter values be z= 2 and = 0.5.Compare a rich parent with w^richP=20 to a poor parent with w^poorP=3.If the parents expect the child to be an actor, what will each parent's choice of patience for the child be? Given patience, whose child (that of the rich or the poor parent) will actually choose to be an actor, and who will be a dentist? Relate your finding to the observation that in many countries, second-generation immigrants have higher earnings compared to first-generation immigrants and natives.