Question

A consumer’s income in the current period is y =100, and income in the future period is y’=120. He or she pays lump-sum taxes t = 20 in the current period and t’=10 in the future period. The real interest rate is 0.1, or 10% per period.
a) Determine the consumer’s lifetime wealth.
b) Suppose that current and future consumption are perfect compliments for the consumer and that he or she always wants to have equal consumption in the current and future periods. Draw the consumer’s indifference curves.
c) Determine what the consumer’s optimal current period and future period consumption are, and what optimal saving is, and show this in a diagram with the consumer’s budget constraint and indifference curves. Is the consumer a lender or a borrower?
d) Now suppose that instead of y = 100, the consumer has y =140. Again, determine optimal consumption in the current and future periods and optimal saving, and show this in a diagram/ Is the consumer a lender or a borrower?
e) Explain the differences in your results between (c) and (d).

Question 2
An employer offers his or her employee the option of shifting x units of income from next year to this year. That is, the option to reduce income next year by x units and increase income this year by x units.
a) Would the employee take this option (use a diagram)?
b) Determine, using a diagram, how this shift in income will affect consumption this year and next year and saving this year. Explain your results.

Question 3
Suppose that the government introduces a tax on interest earnings. That is, borrowers face a real interest rate of r before and after the tax is introduced, but lenders receive an interest rate of (1-x)r on their savings, where x is the tax rate. Therefore, we are looking at the effects of having x increase from zero to some value greater than zero, with r assumed to remain constant.
a) Show the effects of the increase in the tax rate on a consumer’s lifetime budget constraint.
b) How does the increase in the tax rate affect the optimal choice of consumption (in the current and future periods) and saving for the consumer? Show how income and substitution effects matter for your answer, and show how it matters whether the consumer is initially a borrower or a lender.

Question 4
Suppose that there is limited commitment in the credit market, but lenders are uncertain about the value of collateral. Each consumer has a quantity of collateral H, but from the point of view of the lender, there is a probability a that the collateral will be worth p in the future period, and probability (1-a) that the collateral will be worthless in the future period. Suppose that all consumers are identical.
a) Determine the collateral constraint for the consumer, and show the consumer’s lifetime budget constraint in a diagram.
b) How will a decrease in a affect the consumer’s consumption and savings in the current period, and consumption in the future period? Explain your results.

Answer 1

solution- Question-1

a-

Current Income = 100

Current Net Income = 100 - taxes

Future Income = 120

Future Net Income = 120 - taxes

Present value of future Net Income = (120-taxes)/1.1

Lifetime wealth = (100-20) + (120-10)/1.1 = 80+110/1.1 = 80+100 = 180

b)

Since current and future consumptions are perfect complements the indifference curves are L shaped with kink at point where c= c'

c)

Current consumption = c

Future consumption = c'

The budget constrant can be written as:

c + c'/(1+r) = 180

As c and c' are perfect complements, so c = c'

c + c/(1+.1) = 180

c + 0.91c = 180

c = 94.24

And c' = 94.24

Savings = y - t - c = 80-94.24 = -14.24. Hence the consumer is a borrower.

With the lifetime wealth of 180, the consumer can have 180 units of current consumption or 180/1.1 = 163.64 units of future consumption. The equilibrium is achived where the budget line touches the IC.

d)

Y = 140

Lifetime wealth = (140-20) + (120-10)/1.1 = 120+110/1.1 = 120+100 = 220

The budget constrant can be written as:

c + c'/(1+r) = 220

As c and c' are perfect complements, so c = c'

c + c/(1+.1) = 220

c + 0.91c = 220

c = 115.18

And c' = 115.18

Savings = y - t - c = 120-115.18 = 4.82. Hence the consumer is a Lender

With the lifetime wealth of 220, the consumer can have 220 units of current consumption or 220/1.1 = 200 units of future consumption. The equilibrium achieved where the budget line touches the IC.

e)

With the increase in current income the consumer is able to save and thus he is lender when his income increases.