Problem Set 4
Prof. Nordhaus and staff
Economics 122a: Fall 2010
Due: Wed. Oct 27 in class
Problem Set 4 (Solutions)
Please put the name of your section leader and section time on your problem set.
Make sure that the work is your own. You may consult with others, but you may not copy their
answers or use their ideas.
1. Consider the life-cycle model discussed in class. A 20-year-old worker knows that her annual income
will be 90 until the age of 30. After that she will earn 270 until she retires at the age of 60. The worker
expects to live until her 80th birthday. The real interest and discount rates are zero.
a. Assume that the worker can borrow and lend without limit but has zero net worth at death.
Compute consumption expenditures for each year.
Lifetime income:
10 years x 90 = 900
30 years x 270 = 8100
20 years x 0 = 0
----------------------------------------Total
= 9000
Consumption per year:
C = 9000/(80-20)= 150
The worker wants to smooth consumption. Each year she will consume 150.
b. Graph income, consumption, and savings as a function of age.
C,S,Y
income
270
consumption
savings
90
20
40
60
80
age
c. What is the marginal propensity to consume for the worker if she receives an unanticipated
one-time windfall in year 30.
MPC = 1/(80-30) = 1/50 = 0.02
d. Repeat (a) through (c) assuming that the worker cannot borrow.
If the worker cannot borrow she will consume all her income until she is 30. After that she will
smooth consumption.
Consumption per year until the worker is 30: C=90
Consumption per year since the worker is 30: C = 8100/(80-30) = 162
C,S,Y
income
270
consumption
savings
90
20
40
60
80
age
MPC = 1/(80-30) = 1/50
e. Assume that the population distribution is stationary (equal numbers at each age). Calculate the
wealth-income ratio and the net savings rate.
Net savings rate = Total savings / Total income
Without loss of generality, we can assume that there is one person at each age. Then, total
income in the economy is 9,000 and total savings is 0.
Net savings rate = 0
The wealth of person is the accumulation of past savings/dissavings. Assuming that there is
only one person in each age, the total wealth in the economy is 63000.
Wealth -income ratio = 63000/9000=7
2. Assume that James, who obey the Fisher model of consumption, can borrow and lend money at the
interest rate r = 10%. His income is $100 in the first period and $200 in the second.
a. Graph the budget constraint. Be sure you indicate the income level in each period and the slope
of the budget constraint.
C2
200
-1.1
100
C1
b. You observe that James consumes $120 in the first period. Graph an indifference curve
consistent with this consumption. How much will he consume in the second period? How much
does he save in the first period? Graph an indifference curve consistent with this consumption.
S = -20 ; C2 = -20 (1.1) + 200 = 178
C2
200
178
100 120
C1
c. The government announces a tax cut before the election. In the first period, it will transfer $10
to each citizen, but in the second period (after the election) it will impose a tax of $11. Graph
the budget constraint. Again indicate the income level in each period and the slope of the
budget constraint.
C2
The budget constraint remains
the same. Only the endowment
point changes
200
189
e
e'
-1.1
100 110
C1
d. What is the impact of the tax cut and later tax on consumption and saving in each period?
There is no impact on consumption
Savings in period 1 is now -10
e. How do parts c) and d) change if James cannot borrow in the first period? (Redo c and d with
this assumption)
Consumption in period 1 will increase from 100 to 110. Savings will remain zero
C2
200
189
100 110
C1
3. Make the same assumptions as in problem 2 for another individual, Jane.
a. Assume that Jane has target consumption for retirement of 255 because she wants
to travel, see the Taj Mahal, climb K2, and enjoy the good life. Compute Jane’s
consumption and saving in the first period.
C1 = Y1 + Y2/(1+r) - C2/(1+r) = 50
S = Y1-C1 = 100-50 = 50
b.
The government has been taken over by supply-side economists. They have been
told that they must raise incentives to increase saving. So they increase the interest
rate to 20 percent. Calculate the impact of the increase in interest rates on Jane’s
first period saving. Is this paradoxical? Explain the paradoxical result?
With a 20% interest rate savings are 45.83. As indicated in the graph, the
income effect dominates the substitution effect. The increase in the interest
rate will make Jane reduce its savings in period 1.
C2
200
100
C1
4. Does college pay?
Assume that the appropriate discount rate is 10 percent per year and that you live forever. The
price level is constant and = 1.
a. If you graduate from high school at age 18, you will earn $40,000 per year. What is
the present value of your earnings discounted back to age 18?
PV = 40,000/0.1 = 400,000
b.
A fancy education requires four years out of the labor force and costs you $50,000
per year. Upon graduation, you earn $70,000 per year forever. What is the present
value of your earnings minus costs, discounted back to age 18? Is college a good
investment?
PV of Cost = 50,000/(1.1) + 50,000/(1.1)2 + 50,000/(1.1)3 + 50,000/(1.1)4 = 158,493
PV of earnings = ( 1/(1.1)4 )* (70,000/0.1) =478,109
PV (earnings - cost) = 319,616
It' s not a good investment
c.
What is the internal rate of return on college costs (defined as the discount rate for
which the present value of the investment is zero)?
We have to think that not enrolling in college means that you will still earn $40,000
per year. Therefore, the internal rate of return is the value of r that solves:
0 = -90,000/(1+r) - 90,000/(1+r)2 - 90,000/(1+r)3 - 90,000/(1+r)4 + 30,000/(1+r)5 +
30,000/(1+r)6 + 30,000/(1+r)7 + ...
The internal rate of return is approximately 7.5%