I need very specific explanations for each part. If necessary, suppositions might be made but please make these explicit and make sure they are reasonable so that I can follow what you did.
You can suppose a utility function as long as its justified.
Task: Assume the production of childcare for each child takes a fixed total amount of effective time, H, summed over the time of mothers, fathers, and market childcare services. The time of mothers and fathers are perfect substitutes hour for hour, while each hour of market time is equivalent to a<1 hours of parents time. H= hw +hf + ahm, where the subscripts refer to mothers, fathers, or market time. The cost of an hour of market childcare time equals wm. Families differ in the wage rates of mothers and fathers, say each has a set (wf and ww). Suppose each family had the same number of children n.
Problem 1: What determines whether the mother’s, father’s, or market’s time is used in childcare?
Problem 2: Women’s wages are on the average below those of men. What does that say about whether mothers or fathers are more likely to stay home to supply childcare services? Would who stays home be affected by the correlation between the wages of husbands and wives across households? Contrast the case of perfect positive correlation between husbands and wives wage rates with that of perfect negative correlation between these wages. Suppose the government now uses lump sum taxes per household to subsidize market childcare subsidy, so that the cost of market childcare time falls to bwm, with b<1.
Problem 3: Would this lead to substitution of market time for parental time in childcare? Would higher or lower wage families be more likely to substitute toward market time (assume husbands and wives wages are perfectly positively correlated)? What happens to the labor force participation of married women?
Problem 4: What is the effect of this program on the efficiency of the economy? Explain your answer.
Problem 5: Suppose the government now instead allows paid leaves to mothers for one year to care for new born children, where the payment equals the wages they had earned, and those who take leaves are guaranteed the same job when they return at the same wage. The government finances the leaves out of lump sum taxes. Would mothers take these leaves? Are they more likely to be taken by richer or poorer families? Assume a perfect positive correlation in these wages.
Problem 6: What is the effect on labor force participation of such a paid leave program? What is the effect of paid leaves on the demand for market childcare? Do paid childcare and paid leaves have the same effects on labor force participation, and use of parental time to care for children?
Problem 7: Would the effects in part f be different if each employer had to finance the leaves for their own employees? Explain your answer.
Problem 8: If lump sum taxes are used to finance the leaves, what is their effect on efficiency of the economy? Answer this question if employers have to finance the leaves?
Problem 9: What would be the effects of childcare subsidies on fertility; that is, on the number of children that families decide to have? Would the subsidies change fertility more in low or high wage families (assume perfect positive wage correlation)