ECON 305 Tutorial 7 (Week 9)
Questions for today:
Ch.9 Problems 15, 7, 11, 12
MC113
Tutorial slides will be posted Thursday after 10:30am, on
http://www.sfu.ca/~haiyunc/teaching.html.
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ1. Ch.9 Problem 1
A consumer's income in the current period is y = 100, and income in the future
period is y0 = 120. He or she pays lump-sum taxes t = 20 in the current period
and t0 = 10 in the future period. The real interest rate is 0.1 per period.
(a) Determine the consumer's lifetime wealth.
In present value terms, lifetime wealth is
we = y − t +
y0 − t0
120 − 10
= 100 − 20 +
= 180.
1+r
1 + 0.1
(b) Suppose that current and future consumptions are perfect complements for
the consumer and that he or she always wants to have equal consumption in the
current and future periods. Draw the consumer's indierence curves.
Indierence curves for perfect complements is L-shaped
Utility function generating these curves has the form
U (c, c0 ) = min{αc, αc0 },
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
α > 0.
July 2,3, 2014
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AQ1. Ch.9 Problem 1
c0
242
c = c0
198
115.18
110
94.24
E1
U1
80 94.24 120
115.18
H. K. Chen (SFU)
U2
E2
180
AQ1 (c)
AQ1 (d)
AQ1 (e)
ECON 305 Tutorial 7 (Week 9)
220
c
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AQ1. Ch.9 Problem 1
(c) Determine what the consumer's optimal current-period and future-period
consumptions are, and what optimal saving is, and show this in a diagram with
the consumer's budget constraint and indierence curves. Is the consumer a
lender or a borrower?
Intertemporal budget constraint:
we = c + (1 + r)−1 c0
⇒
c0 = 0.91(180 − c).
(1)
Given the utility function, optimality occurs when
c = c0 .
(2)
Solving (1) and (2) for c, c0 , we get c = c0 = 94.24.
Saving in current period is s = y − t − c = −14.24. Therefore the consumer
is a borrower.
Draw
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ1. Ch.9 Problem 1
(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?
Present value wealth is now
we = 140 − 20 +
120 − 10
= 220.
1 + 0.1
Optimality condition (c = c0 ) and budget constraint (c0 = 0.91(220 − c))
together imply
c = c0 = 115.18,
s = 4.82.
The consumer now is a lender.
Draw
(e) Explain the dierences in your results between parts (c) and (d).
Consumer switches from borrower in (c) to lender in (d) due to a suciently
large increase in the current period income.
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ2. Ch.9 Problem 2
An employer oers his or her employee the option of shifting x units of income
from next year to this year. That is, the option is 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)?
Suppose t = t0 = 0. Initial lifetime wealth is
we0 = y + (1 + r)−1 y0 .
Lifetime wealth after the shift is
we1 = y + x + (1 + r)−1 (y0 − x)
= y + (1 + r ) −1 y0 + x 1 − (1 + r ) −1
|
{z
}
|
{z
}
=we0
>0
if r > 0
we1 > we0 whenever r > 0, and so the employee should take the option,
provided that interest rate is positive.
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ2. Ch.9 Problem 2
c0
(1 + r)we1
(1 + r)we0
slope = −(1 + r)
y0
y0 − x
we1
y
H. K. Chen (SFU)
y + x we0
ECON 305 Tutorial 7 (Week 9)
c
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AQ2. Ch.9 Problem 2
(b) Determine, using a diagram, how this shift in income will aect consumption
this year and next year and saving this year. Explain your results.
The shift in income creates a pure income eect (shifts budget constraint
outwards in a parallel fashion).
Thus, both c, c0 will increase.
Saving will increase:
c0 = y0 + (1 + r)s
Since c0 increases while y0 decreases, s must increase for the equality to hold.
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ3. Ch.9 Problem 3
Consider the following eects of an increase in taxes for a consumer.
(a) The consumer's taxes increase by ∆t in the current period. How does this
aect current consumption, future consumption, and current saving?
c0
An increase in tax causes
budget constraint to shift
inwards.
(1 + r)we0
Consumption in both
periods will drop.
Saving also goes down:
(1 + r)we1
E1
y0
y − ∆t
H. K. Chen (SFU)
Consumption smoothing
implies that c will
decrease by an amount
less than ∆t
Recall s = y − t − c
E0
y we1
we0
c
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AQ3. Ch.9 Problem 3
(b) The consumer's taxes increase permanently, increasing by ∆t in the current
and future periods. Using a diagram, determine how this aects current
consumption, future consumption, and current saving. Explain the dierences
between your results here and in part (a).
c0
A permanent increase in tax
shifts budget constraint
inwards.
(1 + r)we0
Consumption in both
periods will drop.
Saving is ambiguous.
E0
y0
E2
y0 − ∆t
y − ∆t
H. K. Chen (SFU)
y
we0
c
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ4. Ch.9 Problem 4
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 eect of having x increase from zero to
some value greater than zero, with r assumed to remain constant.
(a) Show the eects of the increase in the tax rate on the consumer's lifetime
budget constraint.
c0
(1 + r)we
slope = −(1 + r)
(1 + (1 − x )r )y + y0
y0
slope = −(1 + (1 − x)r)
H. K. Chen (SFU)
Budget constraint for
borrowers doesn't change.
E
y
Tax on interest income
causes lender's budget
constraint to pivot down
around initial endowment.
we
c
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ4. Ch.9 Problem 4
(b) How does the increase in the tax rate aect the optimal choice of
consumption (in the current and future periods) and saving for the consumer?
Show how income and substitution eects matter for your answer, and show how
it matters whether the consumer is initially a borrower or a lender.
c0
Consumer is unaected if
initially a borrower.
(1 + r)we
If initially a lender,
(1 + (1 − x )r )y + y0
A
and post-policy
consumption bundle is at
point C,
B
C
y0
then there will be
substitution eect (A → B)
and income eect (B → C).
E
y
H. K. Chen (SFU)
we
c
ECON 305 Tutorial 7 (Week 9)
c0 decreases as a result, but
eect on c (hence s) is
ambiguous and depends on
shape of utility
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AQ5. Ch.9 Problem 5
A consumer's income in the current and future periods are y and y0 , respectively.
He pays taxes t and t0 in the two periods. The consumer can borrow and lend at
the real interest rate r. This consumer faces a constraint on how much he can
borrow. That is, the consumer cannot borrow more than x, where x < we − y + t,
with we denoting lifetime wealth. Use diagrams to determine the eects on the
consumer's current consumption, future consumption, and savings of a change in
x, and explain your results.
c0
Saving is unaected.
There is a borrowing constraint B̄.
(1 + r)we
Budget line becomes vertical at B̄.
y0 − t0
If borrowing constraint is binding,
IC touches budget constraint at the
kink.
E
x
y − t B̄
we
we − y + t
H. K. Chen (SFU)
c
If borrowing constraint not binding,
the usual tangency condition
applies.
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ6. Ch.9 Problem 7
Suppose that all consumers are identical, and also assume that the real interest
rate r is xed. Suppose that the government wants to collect a given amount of
tax revenue R, in present value terms. Assume that the government has two
options: (i) a proportional tax of s per unit of savings, in that the tax collected
per consumer is s(y − c); (ii) a proportional tax u on consumption in the current
and future periods, so that the present value of the toal tax collected per
0
consumer is uc + 1uc
+r . Note that the tax rate s could be positive or negative. For
example if consumers borrow, then s would need to be less than zero for the
government to collect tax revenue. Show that option (ii) is preferable to option (i)
if the government wishes to make consumers as well o as possible, and explain
why this is so. [Hint: Show that the consumption bundle that consumers choose
under option (i) could have been chosen under option (ii), but was not.]
The hint suggests us to appeal to the weak axiom of revealed preference
(WARP).
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ6. Ch.9 Problem 7
Quick review of WARP
c0
Suppose there are two consumption
bundles A and B.
Both are aordable under red budget
constraint, where A is observed to be
chosen.
A
B
B is aordable under blue budget
constraint and observed to be chosen,
while A is not aordable.
c
H. K. Chen (SFU)
We conclude that A must be preferred
to B, because when both A and B are
aordable, A is chosen over B.
ECON 305 Tutorial 7 (Week 9)
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AQ6. Ch.9 Problem 7
In this question, the two tax schemes induce two dierent budget constraints.
Let (c1 , c10 ) be the consumption bundle chosen under scheme that taxes
savings, and (c2 , c20 ) the bundle chosen under scheme that taxes
consumptions.
We need to show that (c1 , c10 ) is aordable under the budget constraint
induced by consumption taxes; and since it's not chosen, it must be inferior
to (c2 , c20 ).
Therefore, the tax scheme that induces (c2 , c20 ) is preferred from the
consumer's perspective.
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ6. Ch.9 Problem 7
Budget constraint under savings tax is
c1 +
c10
y0
= y+
− s ( y − c1 ) .
1+r
1+r
(1)
Budget constraint under consumption taxes is
c20
= y+
1+r
c0
(1 + u) c2 + 2
= y+
1+r
c2 +
⇒
c0
y0
− u c2 + 2
1+r
1+r
(2)
y0
.
1+r
(3)
Since tax revenue is the same under both schemes,
c0
R = s(y − c1 ) = u c2 + 2
.
1+r
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
(4)
July 2,3, 2014
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AQ6. Ch.9 Problem 7
From (4), it follows that
−1
c0
u = R c2 + 2
1+r
−1
c20
c0
c2 + 2
1+r
1+r
−1
y0
y0
= y+
y+
−R
1+r
1+r
⇒
1+u =
eq. (2) & (4)
=======⇒
R + c2 +
Evaluate (c1 , c10 ) in budget constraint under consumption tax (3):
c10
(1 + u) c1 +
Q y+
1+r
−
1
0 c
y0
y0
1
y+
c1 +
= y+
y+ −R
1 + r 1 + r
1 + r
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
y0
1+r
y0
1+r
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AQ6. Ch.9 Problem 7
Therefore, (c1 , c10 ) is exactly aordable under the consumption taxes scheme.
But (c2 , c20 ) is chosen over (c1 , c10 ).
We thus conclude that consumption taxes is preferred to tax on savings.
Despite the mathematics, the intuition is simple:
Tax on savings changes the slope of the budget line, thus creating income and
substitution eects.
The consumption taxes in this question are equivalent to a lump-sum tax,
which creates a pure income eect, and hence lead to no distortion in
consumer's behavior.
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
19 / 24
AQ7. Ch.9 Problem 11
Suppose a consumer has income y in the current period, income y0 in the future
period, and faces proportional taxes on consumption in the current future periods.
There are no lump-sum taxes. That is, if consumption is c in the current period
and c0 in the future period, the consumer pays a tax sc in the current period and
s0 c0 in the future period where s, s0 are tax rates. The government wishes to collect
total tax revenue in the current and future periods, which has a present value of
R. Now suppose that he government reduces s and increases s0 , in such a way that
it continues to collect the same present value of tax revenue R from the consumer,
given the consumer's optimal choices of current- and future-period consumptions.
(a) Write down the lifetime budget constraint of the consumer.
Lifetime budget constraint is
(1 + s)c +
H. K. Chen (SFU)
y0
(1 + s0 )c0
= y+
.
1+r
1+r
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AQ7. Ch.9 Problem 11
(b) Show that lifetime wealth is the same for the consumer, before and after the
change in tax rates.
Lifetime wealth is y + (1 + r)−1 y0 , which is independent of s, s0 .
(c) What eect, if any, does the change in tax rates have on the consumer's
choice of current and future consumptions, and on savings? Does Ricardian
equivalence hold here? Explain why or why not.
The change will in general aect consumption and saving decisions.
From the budget constraint, relative price of consumption is
(1 + r)(1 + s)/(1 + s0 ),
(1 + s)c +
(1 + s0 )c0
y0
= y+
.
1+r
1+r
Changes in s, s0 alters the slope of the budget line, thus creating distortions in
consumer's decisions.
So Ricardian equivalence does not hold here.
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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AQ8. Ch.9 Problem 12
Suppose in our two-period model of the economy that the government, instead of
borrowing in the current period, runs a government loan program. That is, loans
are made to consumers at the market real interest rate r, with the aggregate
quantity of loans made in the current period denoted by L. Government loans are
nanced by lump-sum taxes on consumers in the current period, and we assume
that government spending is zero in the current and future periods. In the future
period, when the government loans are repaid by consumers, the government
rebates this amount as lump-sum transfers (negative taxes) to consumers.
(a) Write down the government's current-period budget constraint and its
future-period budget constraint.
Current period: T = L; future period: T 0 = −(1 + r)L.
(b) Determine the present-value budget constraint of the government.
PV budget constraint:
L + (−L) = T +
H. K. Chen (SFU)
T0
1+r
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AQ8. Ch.9 Problem 12
(c) Write down the lifetime budget constraint of a consumer.
Consumer's budget constraint:
c+
c0
= (y + ` − t) +
1+r
y0 − (1 + r)` − t0
1+r
where the lower case `, t, t0 are loan and taxes for an individual consumer.
(d) Show that the size of the government loan program (i.e., the quantity L) has
no eect on current consumption or future consumption for each individual
consumer and that there is no eect on the equilibrium real interest rate. Explain
this result.
Loan size does not change the PV of taxes, and does not aect interest rate.
Therefore, no change in consumptions.
H. K. Chen (SFU)
ECON 305 Tutorial 7 (Week 9)
July 2,3, 2014
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MC113
MC solutions:
15:
B
C
C
D
A
610:
A
B
B
B
C
1113:
B
B
A
H. K. Chen (SFU)
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July 2,3, 2014
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