Download Due Date: Friday, September 17th

Problem Set 5
FE312 Fall 2008
Rahman
Some Answers
Chapter 16:
1) Use the Fisher model discussed in 16-2 of the text to analyze Jack’s consumption
behavior. Say Jack actually borrows income to increase his first-period consumption.
a) Draw Jack’s budget line and indifference curve to illustrate this case. Be sure to
label his income profile (Y1 and Y2) and his consumption profile (C1 and C2).
Y2
C2
Y1
C1
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Problem Set 5
FE312 Fall 2008
Rahman
b) Now say there is a higher interest rate. Illustrate the change in your diagram. Be
sure to label the new budget line and his new consumption profile (C1,new and
C2,new).
Y2
C2,new
C2
Y1 C1,new C1
We can break the effect on consumption from this change into an income effect and a
substitution effect. The income effect is the change in consumption that results from the
movement to a different indifference curve. Because the consumer is a borrower, the
increase in the interest rate makes the consumer worse off – that is, he or she cannot
achieve as high an indifference curve. If consumption in each period is a normal good,
this tends to reduce both C1 and C2.
The substitution effect is the change in consumption that results from the change in the
relative price of consumption in the two periods. The increase in the interest rate makes
second period consumption relatively less expensive; this tends to make the consumer
choose more consumption in the second period and less consumption in the first period.
On net, we find that for a borrower, first period consumption falls unambiguously when
the real interest rate rises, since both the income and substitution effects push in the same
direction. Second period might rise or fall, depending on which effect is stronger. In the
figure above, I show the case in which the substitution effect is stronger than the income
effect, so that C2 increases.
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Problem Set 5
FE312 Fall 2008
Rahman
2) Again consider the Fisher model of 16-2, but now let’s use some actual numbers
(Note: This one’s tougher [and therefore more fun], and will require a bit of
calculus – keep in mind that optimization requires that the marginal rate of
substitution equals the slope of the budget line).
Suppose that Bart has the following utility function: U  C1C 2  , where C1 and C2 are
Bart’s levels of consumption in Periods 1 and 2, respectively. Suppose that his income is
$120 in Period 1 and $100 in Period 2, and the real interest rate is 25 percent.
1/ 2
a) Calculate Bart’s optimal levels of consumption in Periods 1 and 2 and his
savings, if any, in Period 1.
First, calculate the marginal utility of C1 and C2:
MU C1  1 / 2C1
1/ 2
MU C 2  1 / 2C1 C2
1/ 2
1// 2
C2
1 // 2
Recall that the optimal condition for Bart is when the slope of the indifference
curve (the marginal rate of substitution) equals the slope of the inter-temporal
budget line. This equation is:
MU C1
 1  r , where the left hand side describes the slope of the indifference
MU C 2
curve (which changes as C1 and as C2 change), and the right hand side is the
slope of the inter-temporal budget line (which is just a constant).
The other equation you need is the inter-temporal budget constraint itself. This is
C1 
C2
Y
 Y1  2
1 r
1 r
So we have two equations and two unknowns, C1 and as C2.
I get C1 = 100 and C2 = 125
This means that Bart is a lender – he has extra income during the first period of
his life to lend out.
b) Suppose that the real interest rate increases to 50 percent. Calculate Bart’s new
optimal levels of consumption in Periods 1 and 2 and his savings (if any) in
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Problem Set 5
FE312 Fall 2008
Rahman
Period 1.
I get C1 = 90 and C2 = 135. This means that Bart has increased his savings.
Chapter 3 (3-3 & 3-4):
3) Soon after his election in 1992, President Clinton proposed to reduce government
spending and increase taxes.
a) What effect would his have on the government budget deficit? What would be the
long term effects on private savings, public savings, and national savings?
We can write national savings as S = (Y – T – C) + (T –G). The first term is
private savings, and the second term is public savings. It is clear that when G
goes down and T goes up, public savings must rise.
What about private savings? The increase in taxes reduces disposable income,
which lowers private savings, but it also lowers consumption (since consumption
is a function of disposable income), which tends to increase private savings.
Which effect dominates? Recall the consumption function, illustrated in figure 36. So long as the marginal propensity to consume is less than one (MPC < 1),
consumption will fall less than the tax increase (in absolute values). Makes
sense? The net result is that private savings will slightly fall.
The overall effect however must be that national savings rises.
b) Use the I,S diagram to state and illustrate what the long-run impact of this
program would be on national saving, investment, and the real interest.
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Problem Set 5
FE312 Fall 2008
Rahman
r
S1
S2
r1
r2
I
S,I
4) Many Congressional Republicans have suggested cutting both government purchase
and taxes.
a) If both taxes and government purchases were cut by equal amounts, state and
explain what the long term effects of this policy would be on private savings,
public savings, and national savings.
Again, we can write national savings as S = (Y – T – C) + (T –G). Clearly,
public savings remains unaffected (since T and G fall by the same amount).
However, private savings ends up actually rising a bit. How? Falling taxes
means there is more disposable income, which allows consumption to rise. But
because the MPC < 1, this increase is less than the actual amount of the tax
decrease. So the term (Y – T – C) rises overall.
Note one of the end lessons in this. A tax and spend policy by the government
ends up crowding out overall savings in the economy, even if the government
maintains a balanced budget! Less government spending and less taxation is
better for the economy in the sense of increasing savings and lowering the cost of
investment.
b) Use the I,S diagram to state and illustrate what the long-run impact of this
program would be on national saving, investment, and the real interest.
Page 5 of 6
Problem Set 5
FE312 Fall 2008
Rahman
Same basic diagram as problem 3b.
5) How would the I,S diagram (Figure 3-8) change if consumption depended on both
disposable income and on the interest rate? More specifically, how would this alter the
conclusions reached in this chapter about the effects of fiscal policy? Draw a new
diagram where consumption is a function of the interest rate, and illustrate the effects of
an increase in government spending, as part of your answer.
Because the interest rate is the return to saving (as well as the cost to borrowing), a
higher interest rate might reduce consumption and increase savings. If so, the saving
schedule would be upward sloping rather than vertical. (This is the case illustrated
in Figure 3-12). Note however from our discussion from Chapter 16 that this would
happen only if lenders have stronger substitution effects rather than income effects
from an interest rate increase. This case is illustrated in Figure 16-7. The lender is
willing to forgo consumption now for a lot more consumption later.
However, it is just as conceivable to have a NEGATIVELY sloped saving scheduled,
where interest rate increases would actually decrease savings. This could happen if
lenders have stronger income effects than substitution effects from an interest rate
increase. (This case is not illustrated in the book, or even discussed). The lender
cares very much about SMOOTHING his or her consumption lifetime profile – the
rise in the interest rate will allow the lender to consume both more now and later;
consequently the lender will cut back on savings.
Finally, note that this case should not change our analysis with respect to changes in
government spending. An increase in government spending shifts the savings curve
to the left and raises the interest rate. This leaves the economy with less overall
savings and investment, same as our traditional case.
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