# Finance assignment question 7

## Exercise 1 (10 points)

Families with capital income might respond differently to changes in payroll taxes than families with income losses.

Suppose a consumer has a utility function U(C,N) = C N where C is weekly consumption measured in dollars (P=SI) and N is weekly leisure time measured in hours.

- Write the expression of the consumer budget line if he each weeks he earns A of non-labor income.
- Using the Lagrangian method find the labor supply function of this consumer. If offered a higher wage, does this consumer work longer or shorter hours?
- In a diagram, draw the consumer's budget line and optimal bundle when T = ISO; A = $2,000 and w = $40.
- In the same diagram, illustrate the consumer optimal bundle when his hourly wage increases to $60. Now suppose the consumer owes money to a bank and has to make weekly payments of D.
- Write the expression of the consumer's budget line.
- Find the labor supply function of this consumer. If offered a higher wage, does this consumer work longer or shorter hours?
- In new diagram, draw the consumer's budget line and optimal bundle when T = 150; D; $2,000 and w = $40.
- In the diagram above, illustrate the consumer optimal bundle when his hourly wage increases to $60.

## Exercise 2 (4 points)

Who is more likely to work, a single woman with no children or a married mother of three? Discuss.

## Exercise 3 (8 points)

Each week, a college student has a time endowment of 21 hours she can use to work for pay or to rest and have fun with her friends. The student's utility function is U(C,N) = CN^{2} where C is earned income measured in dollars and N is time spent on unpaid activities measured in hours. Jobs on campus pay $13.31 an hour, while jobs off-campus pay $17.28 an hour.

- Write the expression of the student's budget line if she works on campus and then the expression of the student's budget line if she works off-campus and must commute Tc hours each week.
- If she chooses to work on campus, how many hours does the student work each week (hint: find her optimal bundle when T-21, w = $13.31)
- In an indifference curve diagram, illustrate the student's budget line and optimal choice if she works on campus.
- Find the expression of the student indirect utility function.
- Use the indirect utility function to find how much time the student is willing to spend commuting to a job off-campus (hint: you are looking for the CV of the wage rate offered by jobs off-campus).
- In the diagram, illustrate this amount of time.

## Exercise 4 (8 points)

A proposal for reforming the U.S. welfare system called for a Negative Income Tax (NIT). Under a NIT, each person is entitled a grant of G dollars per month. For every dollar the person earns, the grant is reduced by t dollars. Suppose government enacts a NIT with a monthly grant of $400 and a flat tax rate is 25%.

- Consider an individual whose hourly wage is $10. Sketch the budget constraint before and after the introduction of the NIT assuming the individual has no tax liabilities otherwise.
- How would the NIT affect the labor supply of a full time worker (i.e. an individual who works 160 hours each month)? Does your answer depend on whether the individual considers leisure time a normal or an inferior good?
- How would the NIT affect the labor supply of a part-time worker (i.e. an individual who works go hours each month)? Does your answer depend on whether the individual considers leisure time a normal or an inferior good? At current legislation, low-income workers with children qualify for the Earned Income Credit (EIC). Under the EIC, parents with two children receive 40 cents for every dollar they earn up to a plateau of $400 per month.
- Consider an individual with two children whose hourly wage is $20. Sketch the budget constraint before and after the introduction of the EIC.
- How does the EIC affect the labor supply if the individual is a full time worker (i.e. works 160 hours each month)? Does your answer depend on whether the individual considers leisure time a normal or an inferior good?
- f) How does the EIC affect the labor supply if the individual is a part time worker (i.e. works 80 hours each month)? Does your answer depend on whether the individual considers leisure time a normal or an inferior good?
- g) Compare how the two income maintenance programs affect income distribution and the efficiency of labor markets. First compare their generosity to part-time and full-time workers, then compare their effect on the labor supply of part-time and full time workers.

## Exercise 5 (6 points)

Empirical analysis suggests that labor supply curves typically slope up when wages are low and slope down when wages are high.

This is sometimes referred to as a backward-bending labor supply.

Suppose that an individual's tastes over consumption and leisure time are described by a constant elasticity of substitution utility function U(C,N) = (0.5C^{-p} + 0.5N^{-p})^{-1/p}

- Derive the labor supply function assuming a time endowment T.
- Illustrate for which values of p the function is increasing in wages and for which it is decreasing.
- Is it possible for the backward-bending labor supply curve to emerge from tastes captured by a CES utility function? For practical purposes, economists only need to worry about modeling tastes accurately at the margin, i.e. around the current bundles that individuals are consuming. This is because low wage workers, for example, might experience wage increases but not so much that they are suddenly high wage workers, and vice versa.
- If you were modeling worker behavior for a group of workers and you modeled each worker's tastes as CES over consumption and leisure time, how would you assume p differs for low-wage and high-wage workers?

## Exercise 6 (4 points)

The government levies a 30% wage tax on Cleopatra, It uses the tax money to finance a parade, The parade's value to Cleopatra is just sufficient to make her as well off as she was before the tax was levied. What is the effect of the government tax and expenditure package on Cleopatra's labor supply? Explain.

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