Questions 1: A tax on earnings
This problem focuses on the labor supply e↵ects of taxes. Assume there is a worker (Cosmo)who has a utility function over money income m and leisure l is u(m, l) = (ml)1/2. Cosmo chooses how many hours to supply to the labor market where h = 16 -l subject to the market hourly wage w. The feasibility constraint is such that neither leisure nor labor supply can exceed 16 hours and cannot be negative. Cosmo has unearned income a = 8.
1. It can be shown (with calculus) that if utility is u(m, l) = (ml)1/2, the marginal utility of leisure is MU(l) = 1/2(m/l )1/2 and MU(m) = 1/2(l/m)1/2.Assuming that the interior solution for labor supply will dominate the corner solutions, what is Cosmo’s optimal labor supply as a function of the market wage w?
2. Now suppose the government levies a positive tax of t on each dollar that Cosmo earns. That is, Cosmo’s after tax wage becomes w(1-t). What is the e↵ect of this tax on Cosmo’s budget constraint? Draw a graph showing both the new and old budgetconstraints (for any positive t).
3. Assuming that the interior solution for labor supply will dominate the corner solutions, NOW what is Cosmo’s optimal labor supply as a function of the market wage w and the tax rate t?
4. Does the number of hours Cosmo supplies to the labor market increase or decrease as t gets bigger (closer to 1)? Based on this, when the tax rate increases, does the income or substitution effect dominate?
Question 2: Welfare Programs and Work Incentives
You will likely need a calculator to do this problem. Mike’s utility for money income (M) and leisure (L) is U(M,L) = M1/3L. There are 16 hours in a day that can be dedicated to work or leisure. Mike earns $12 per hour. Assume Mike has no unearned income.
Given Mike’s utility function, it can be shown that his marginal utility of leisure is MUL =M1/3 and his marginal utility of money income is MUM = L / 3M2/3 .
1. What is Mike’s marginal rate of substitution between leisure and money income (MRSL,C)?
2. Assuming Mike is at an interior optimal point, what is his optimal amount of money income and leisure per day? How many hours does he choose to work per day?
3. We can be sure that Mike does indeed choose to be at an interior point (that is, working a positive amount and taking a positive amount of leisure) by comparing his utility at the interior optimal point with his utility at each corner point. What is Mike’s utility at the interior optimal point in Part #2? What is his utility if he works all 16 hours a day (so that L = 0)? What is his utility if he does not work at all (so that L = 16)? As a result, are we sure that Mike will choose an interior point?
4. Now suppose that Mike’s wage increases to $18 per hour. Now what is his optimal amount of money income and leisure per day? How many hours does he choose to work per day? When Mike’s wage changes from $12 to $18, does the income or substitution effect dominate?
5. Continue to suppose that Mike’s wage is $18 per hour. Suppose that the government starts a welfare policy that pays $B per day to non-workers (and pays $0 per day to workers). At what value of B will Mike opt out of the labor force in order to go on welfare? (Hint: Mike ranks any two possible combinations of M and L by comparing the utility levels associated with each alternative).
6. (tough) Suppose the government sets B = 50. Will Mike work at $18 per hour? Now what is the lowest wage that will induce Mike to go back into the labor market? (Hint: this is the wage at which, if Mike works, he gets the same utility as when he is on welfare.)
Question 3. We can be sure that Mike does indeed choose to be at an interior point (that is, working a positive amount and taking a positive amount of leisure) by comparing his utility at the interior optimal point with his utility at each corner point. What is Mike’s utility at the interior optimal point in Part #2? What is his utility if he works all 16 hours a day (so that L = 0)? What is his utility if he does not work at all (so that L = 16)?
As a result, are we sure that Mike will choose an interior point?
Question 3: Child Care and Female Labor Supply
The presence of pre-school children often complicates women’s labor supply decisions by forcing them to consider the costs of child care. In this problem, you are asked to consider the labor supply decision of a single mother with one child of pre-school age.
1. Show (graphically) how the woman’s labor supply decisions are a↵ected by child care costs. (Hint: For every hour that a woman is working and hence cannot look after her child, she must purchase an hour of child care services in the market, at price c).
What is the effect of child care costs on the labor force participation rate of women with young children?
2. The government is considering two proposals designed to assist women in covering the costs of child care:
Proposal A: A subsidy equal to some fraction of a household's total child care expense.
Proposal B: A child subsidy to households for each child of pre-school age.If the govenment wishes to increase the labour supply and labour force participation of single mothers which of these two policies would you recomment and why?
Question 4: Child Labor
Suppose that it is well known that many children work in low-skill jobs in poor country “X.” If rich country “Y” buys goods from poor country “X,” then the increase in demand for their products is large enough to raise the wages of all low-skill workers in country “X.” Because of this, a politician from country “Y” makes the following statement:
“We should not buy any goods from country “X.” If we do, it will only increase the number of child laborers there.”
1. Analyze the politician’s statement using economic theory. In doing so, suppose that parents make the decisions about whether their children will work or do other things (e.g. go to school). (Hint: your answer should include something about income and substitution e↵ects.)
2. Is there any evidence from the real world that suggests that as wages continue to rise, child labor eventually declines?