ECON 304
HOMEWORK # 4
due: June 7 in class
1. A consumer has current income (y1 ) of $20 and future income (y2 ) of $100. The real interest rate is 10% per
period.
a. Find the consumer’s present value of income
Let PV denote the present value of income for this consumer. Then:
P V = y1 +
y2
100
= 20 +
= 110.909
1+r
1.1
b. Write the equation for the consumer’s budget constraint, and graph it
Remember we can rewrite the budget constraint as follows:
c2 = [y1 (1 + r) + y2 ] − (1 + r)c1
So, if we plug in the values for y1 , y2 and r. We get:
c2 = 122 − 1.1c1
The budget constraint looks like this:
1
c. Suppose that the consumer’s goal is to smooth consumption completely. That is, he wants to have the same
level of consumption in both period 1 and 2. How much will he consume and save in period 1?
Since the individual wants to smooth consumption in both periods, we know that c1 = c2 = c. We can
use the budget constraint of the individual to find the value of c:
c1 +
y2
c2
= y1 +
= 110.909
1+r
1+r
Since c1 = c2 = c we can plug this into the budget constraint
c
= 110.909
1.1
1.909c = 110.909
110.909
= 58.09
c=
1.909
c+
Hence, the individual will consume 58.09 in both periods. This implies that his saving in period 1 is:
s1 = y1 − c1 = 20 − 58.09 = −38.09
This individual is borrowing $38.09 to smooth his consumption across time.
2
d. How will his consumption and saving in period 1 be affected if y1 increases to $31?
Repeat the same operation as above, but using the new present value of income 31+ 100
1.1 = 121.909. This im121.909
plies that c = 1.909 = 63.86. The new saving in period 1 is going to be s1 = y1 −c1 = 31−63.86 = −32.86.
So we can see that even though consumption in period 1 goes up, saving increases as well, in the sense
that this individual is borrowing less money than when his PV of income was $110.909.
e. How will his consumption and saving in period 1 be affected if y2 increases to $111?
Repeat the same operation as above, but using the new present value of income 20+ 111
1.1 = 120.909. This implies that c = 120.909
=
63.33.
The
new
saving
in
period
1
is
going
to
be
s
=
y
−c
1
1
1 = 20−63.33 = −43.33.
1.909
There are two things to notice here: (1) the increase in $11 in y1 increases the PV of income more than
the increase in $11 in y2 . In other words, one extra dollar today, is more valuable than one extra dollar
tomorrow. (2) In this case, since income in period 1 stays constant and consumption in period one goes
up, savings in period 1 will fall.
f. How will his consumption and saving in period 1 be affected if the interest rate falls to 5%? Optimal
consumption in this case is given by c[2 + r/1 + r] = y1 + y2 /1 + r, so it obtains c = 59.02 and s1 = −39.02.
From the substitution effect we see that consumption in period 1 increases from 58.09 to 59.02 because
current consumption has become cheaper. The income effect goes in the same direction (increasing c1 ).
Since the individual borrows money in period 1, the fall in r makes him financially better off (he has to
pay less interest on his loan in period 2).
2. An economy has full-employment output of 9,000, and government purchases are 2,000. Desired consumption
and investment are as follows:
r
Cd
Id
2
3
4
5
6
6,100
6,000
5,900
5,800
5,700
1,500
1,400
1,300
1,200
1,100
a. Find desired national saving (S d ) for each value of the real interest rate
Remember that S d = Y − C d − G, so we can complete the table above:
3
r
Cd
Id
Sd
2
3
4
5
6
6,100
6,000
5,900
5,800
5,700
1,500
1,400
1,300
1,200
1,100
900
1,000
1,100
1,200
1,300
b. If the goods market is in equilibrium, what are the values of the real interest rate, desired national saving
and desired investment?
Equilibrium in the goods market is achieved when S d = I d . From the table in part (a), we see that the
equilibrium real interest rate is 5% and equilibrium investment and saving equal $1,200
3. Numerical exercise # 6, Chapter 4, page 152, Abel & Bernanke
a. Find an equation relating desired national saving, S d to r and Y
Sd = Y − Cd − G
S d = Y − [3, 600 − 2, 000r + 0.1Y ] − 1, 200
S d = 0.9Y − 4, 800 + 2, 000r
b. Find the equilibrium real interest rate
Plugging Y = 6, 000 in the equation in part (a) we obtain S d = 600 + 2, 000r. Equilibrium real interest is
solved from:
Sd = I d
600 + 2, 000r = 1, 200 − 4, 000r
6, 000r = 600
r = 0.1 = 10%
c. Government purchases rise to 1,400. How does this increase change the equation describing desired national
saving? Show the change graphically. What happens to the equilibrium real interest rate
4
The equation for desired saving becomes
S d = 0.9Y − 5, 040 + 2, 000rS d = 360 + 2, 000r
The equilibrium interest rate can be solved for as in part (b)
Sd = I d
360 + 2, 000r = 1, 200 − 4, 000r
6, 000r = 840
r = 0.14 = 14%
The increase in G reduces national saving, shifting the S d curve to the left and increasing the real interest rate
Sd1
r
360
600
Sd
Sd
5