solve ASAP please give correct answer

Problem 6: The rotten kid theorem In a threeperson household (father, mother, child) the parents not only care about their own well-being, but also about the happiness of their child. The child, on the other hand, is completely egoistic. With its work effort the child can increase the famin income. An effort a mereases the family income by 20:: whereas the child incurs a disutility of (.12. The parents observe the child's effort and decide about the child's pocket mone)r of amount b. The child's preferences are given by the utility function Uc(a.b) = h1(100 a2 + b). The parents\" utility function is Vp = 111(400 + 20a b) + Uc'(a. b). I- W'rite this game in an extensive form. 2. Show that in the unique subgameperfect equilibrium the egoistic child Chooses its effort a such that the family income minus the child's disutility.r is maximized (this is the rotten kid theorem). Note that the total family income is 100+400+20L [Hintz Solve for the equilibrium value of a and compare it with the value of a maximizing the family income minus the child's disutility]. 3- Now assume that the parents are. completely egoistic: Vp = ln(100 + We - b). Find the aubgameperfect equilibrium. Does the equilibrium effort a still maximize the family income minus the child's disutilit}r of effort