# Consider a single mother with the utility function U = 2/3 log(x) + 1/3 log( ), where x is…

Consider a single mother with the utility function U = 2/3 log(x) + 1/3 log(), where x is consumption and  is leisure. The mother can work up to 100 hours per month. Any of the 100 hours that are not worked are leisure hours. She earns a wage of \$10 per hour and pays no taxes. The consumption price is normalized to \$1. To be able to work, she has to incur a child care cost of \$5 for every hour worked.

a. Suppose that there is no tax and welfare benefits. How many hours will she work and what will be her consumption level? Draw the graph depicting her budget set with consumption on the vertical axis and leisure on the horizontal axis.

b. Suppose that the government introduces a negative income tax (NIT) that guarantees an income of \$200 per month. The benefit is taken away one for one as earnings increase. Draw the new budget set. Compute the new number of hours worked and consumption level. Has consumption increased and is the mother better off? Why or why not?

c. Now suppose the income guarantee is reduced by one-half to the amount of \$100 per month. What is the new number of hours worked and the consumption level? Compare with your result in part b.

d. Now consider again the income guarantee in part b of \$200 per month, and suppose that the government complements this benefit by offering free child care. Draw the new budget set and calculate the number of hours worked and consumption level. Calculate the total cost of the program for the government. How does it compare with the program in part b? Define program exigency as the ratio of the mother’s consumption to government expenditure. Which program dominates on exigency grounds?