Please help with the whole question. The topic is on Child Investment.

2. The parent in a single-parent household has a utility function given by u = 02an + 0.3lnc + 0.5lnk, where l is leisure, c is consumption of a good purchased in the market, and k is the cognitive ability of the child. The production function of child quality is given by k = T0'560'4k8'6 where 160 is initial child quality, 6 is the money spent on investment in the child, and 'r is time spent by the mother investing in the child. Let k0 = 10. The mother has a wage of w = 5, nonlabor income of y = 20, and a time endowment of T = 20. 1. Find her utility-maximizing decisions h,*, T*, and 6*. What is nal child qual- ity It? What is the mother's utility? 2. The government sends all households with a young child a check for 40 dollars. What is the effect on k and the mother's utility? 3. The government sends all households with a young child investment goods for the child that have a value of 40. These goods can only be used for child investment, meaning that e Z 40. Find the mother's utility-maximizing choices in this case, and nd the child's cognitive ability value, k. Find the mother's utility level in this case. 4. Compare the answers to parts 2 and 3. Explain the difference in the results