ECON_481_Exam_2_SPRING_2023_PRACTICE_SOLUTIONS
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Subject
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Date
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Econ 481: Spring 2023 Exam #2_SOLUTIONS
June 3, 2023
Please choose the part that you are more comfortable with.
Part A
Problem #1
Consider a project that is to yield
10
at the end of each period - starting from period
1
- in
perpetuity, but requires investment of
105
at the beginning of period
1
and the beginning of period
2
.
Would you recommend implementing such a project?
a) Yes, as long as the interest rate is higher than
10%
, b) Yes, as long as the interest rate is less
than
5%
, c) Yes, as long as the interest rate is less than
10%
, d) None of the above.
D
Problem #2
Consider two potential public projects. The first one leads to a net benefit of
100
and allows
the two citizens to attain a level of happiness
U
1
= 10
and
U
2
= 12
.
The other project leads to a
net benefit of
120
,
and allows for the two citizens to enjoy utility levels of
U
1
= 9
and
U
2
= 11
.
Which of the two projects is likely to be implemented if traditional cost benefit analysis is applied?
Which of the two should be implemented by if welfare is to be used as a criterion?
a) In both cases the first, b) The first wins in terms of efficiency the second in terms of welfare,
c) The second wins in terms of efficiency and the first in terms of welfare, d) In both cases the
second.
C
Problem #3
Imagine that the US decides to build a wall on the border with Mexico. Construction demands
in total
1
mln hours of labor. The US is considering three options to construct the Wall. Option
one involves hiring only Americans, option two allows for employment of Mexican citizens in the
process of construction, and options three allows for employment of Mexicans, but only if they
happen to be compensated by the government of Mexico. Which of the three options is the worst
from the US perspective?
a) The third, b) The second, c) The first, d) All three are equally good.
C
1
Problem #4
Imagine that there are in total
2
N
+ 1
individuals in the economy. The preferences of a single
individual
i
∈ {
1
,
2
,
3
, ...,
2
N
+ 1
}
are represented with
U
i
=
-
2(
G
-
i
)
2
,
where
G
denotes the level
of provision of a given public good. In a two party system with a simple majority voting rule what
type level of
G
can we expect to be chosen?
a)
1
, b)
N
+1
2
, c)
N
+ 1
, d) None of the above as preferences are not single peaked.
C
Problem #5
Continue with the assumptions of the previous problem, but now assume the government is
interested in maximizing welfare represented with
W
=
U
1
+
U
2
+
U
3
.
What level of
G
do you
expect to see in this case?
a)
1
, b)
N
+ 1
, c)
2
, d) None of the above.
C
Problem #6
There are three families in a given community. Their preferences over the quality of eduction
are given by
H
1
M
1
L
for family
1
M
2
L
2
H
for family
2
and
L
3
M
3
H
for family
3
,
where
H
denotes education of high quality,
M
education of
medium quality, and
L
education of low quality. Which option is going to be chosen if residents
decide on their choice with a simple majority voting rule?
a) L, b) M, c) H, d) The rule does not lead to a conclusion as group preferences induced with
voting are circular.
B
Problem #7
Consider three individuals -
A, B,
and
C
- with private marginal benefits of a given public good
given with
P
A
= 100
-
2
Q, P
B
= 150
-
3
Q,
and
P
C
= 50
-
Q.
It turns out that in this economy it
is optimal to provide
40
units of the good. What is the marginal cost of delivery of the good?
a)
60
, b)
40
, c)
50
, d) None of the above.
A
Problem #8
Continue with the assumptions of the previous problem.
How much should individual
A
be
charged to enjoy the good?
a)
30
, b)
10
,
c)
20
, d) None of the above.
C
Problem #9
Imagine that in a given community,
A,
individuals value education according to
P
A
= 120
-
3
E
A
and in a neighboring community,
B,
the valuation is given with
P
B
= 80
-
2
E
B
.
Imagine that the
marginal cost of education is the same in both places and is equal to
30
.
What is the level of
education chosen in each place?
a)
E
A
= 40
and
E
B
= 40
, b)
E
A
=
E
B
= 30
, c)
E
A
= 30
,
and
E
B
= 25
, d) None of the above.
C
2
Problem #10
Would you agree that a policy of subsidizing the provision of education in community
B
and
of taxing the provision of education in community
A
can lead to equalization of consumption of
education, which in turn must be welfare improving?
a) There is no need to intervene in this manner as the levels of education chosen are already
equal, b) It would lead to equalization, but it need not be welfare improving, c) It would further
increases the discrepancy in the level of education in both places, d) Would bring the two levels of
education closer and would definitely lead to a higher welfare in both places.
B
Problem #11
Imagine that individual preferences are represented with
U
(
C
) =
√
C
and that there are two
states of nature
L
and
H
with consumption in the low state
C
L
being equal to
1
and in the high
state
C
H
being equal to
4
.
Furthermore, assume that insurance companies are run efficiently and
operate in a competitive environment. Consider an individual who ends up in the low state with
probability
p
L
.
What is the premium - per dollar of coverage - that this individual will be charged
for complete insurance?
a)
1
-
p
L
, b)
3
, c)
p
L
, d)
0
.
C
Problem #12
Continue with the assumptions of the previous problem, but now assume that the insurance
companies are able to earn profit. What is the highest premium - per dollar of coverage - that the
individual is willing to pay?
a) Same as in the previous problem, b)
p
L
(1+
p
H
3
), c)
4
, d) None of the above.
B
Problem #13
Now imagine that there are two categories of individuals. In the first
p
2
L
=
1
4
and in the second
p
2
L
=
3
4
.
Imagine that the premium is equal to
1
2
per dollar of insurance. Who will be insurance in
this case?
a) Both, b) Only the second, c) Only the first, d) Neither.
C
Problem #14
Consider an unfunded social security system.
Assume that individuals at time
t
contribute
fraction
τ
t
= 0
.
15
of their wages to the system, and individuals at time
t
+ 1
are expected to
contribute fraction
τ
t
+1
= 0
.
3
of their wages.
Furthermore, assume that wages are expected to
increase at
2%
per period and that labor force expands at
1%
per period.
Finally, assume that
the return on investment in physical capital is equal to
3%
.
What is the rate of return from -
approximately - participating in such a system?
a)
106%
, b)
103%
, c)
102%
, d) None of the above
3
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Problem #15
Continue with the assumptions of the previous problem, but now assume that we are dealing
with a fully funded system. What is the return in this case?
a)
6%
, b)
3%
, c)
2%
, d)
1%
.
B
Problem #16
Which of the following goods should be provided locally (city, county level ?)
a) Elementary School, b) Research Hospital, c) Intercontinental airport, d) A welfare program
for families in need.
A
Problem #17
Which of the following goods should be provided at the federal level?
a) Public transportation, b) K-12 education, c) Street lights, d) National Defense.
D
Problem #18
In which of the following self-insurance is most likely to be effective?
a) Social security insurance, b) Automobile insurance, c) Unemployment insurance, d) Disability
insurance.
A
Problem #19
Imagine that the federal government becomes concerned with the the ability to read by elemen-
tary school students and decides to encourage reading by young people.The government has two
options. It can either offer a matching grant (one for one) or a block grant of equal value. Which
of the two options should the government resort to given its stated objective.
a) Block grant as long as it is smaller than the original value of spending on books b) Matching
grant, c) Always a block grant, d) The two must be equally effective as both happen to be of the
same value.
B
Problem #20
Would you recommend provision of complete insurance in the context unemployment insurance?
a) No, as moral hazard issues are likely to become relevant, b) Only for mid-aged individuals,
c) Yes, as self-insurance is unlikely to be relevant in this case, d) Yes, as it must definitely lead to
an improvement in the quality of matches between employees and employers.
A
4
Part B
Problem #1
The city of Metropolita added a new subway station in a neighborhood between two existing
stations. After the station was built, the average house price increased by
$10
,
000
and the average
commute time fell by
15
minutes per day. Suppose that there is one commuter per household, that
the average commuter works
5
days per week,
50
weeks per year, and that the benefits of reduced
commuting time apply to current and future residents forever.
Assume an interest rate of
5%
.
Produce an estimate of the average value of time for commuters based on this information.
Letting
V
denote the value of an hour, the annual value of the time savings from the change can
be written as (
5
days/week
×
50
weeks/year
×
0
.
25
hours/day)
×
V
, or
62
.
5
V
per year. The present
value of
62
.
5
V
per year forever at a
5%
discount rate is
62
.
5
V/
0
.
05
. The implication of the change
in house prices is that households were willing to pay
$10
,
000
to gain
15
minutes per commuting
day, that is, that the present value of the time savings is
$10
,
000
. Setting
62
.
5
V/
0
.
05 = $10000
,
and solving gives
V
= $8
.
5
Problem #2
Imagine a family with preferences represented with
U
(
C, E
) = (1
-
a
)
log
(
C
) +
alog
(
E
)
,
where
a
is a parameter,
C
denotes the level of consumption and
E
denotes the level of education.
Assume that the price of consumption is equal to
1
and the price of educations is equal to
1
as well.
Finally, let the level of income of a given family be equal to
Y
= 100000
.
i) How does the choice of
E
depend on
a
?
At the optimal point we have
MU
C
MU
E
=
P
C
P
E
=
1
1
= 1
,
i.e.,
MU
C
=
MU
E
,
which leads to
1
-
a
C
=
a
E
and in turn to
(1
-
a
)
E
=
aC.
Recall that the budget constraint is
C
+
E
= 100000
.
Combining
the two we obtain
C
= (1
-
a
)100000
and
E
=
a
100000
.
ii) Imagine that the government chooses to make public schools available for free to all residents.
Assume that public education is worth 20000 and is financed from other sources. Determine the
values of
a
that lead to a choice of public education.
Agents who choose not to go to public schools attain utility
U
P R
= (1
-
a
)
log
(
C
)+
alog
(
E
) = (1
-
a
)
log
((1
-
a
)100000)+
alog
(
a
100000) =
log
(1
-
a
)
1
-
a
+
loga
a
+
log
(100000)
,
and those who choose to go to public schools choose
C
= 100000
and
E
= 20000
attain utility of
U
P U
= (1
-
a
)
log
(100000) +
alog
(20000) =
log
(100000)
-
alog
(5)
.
Individual will choose public education over private schools when
U
P U
> U
P R
,
i.e., when
log
(100000)
-
alog
(5)
> log
(1
-
a
)
1
-
a
+
loga
a
+
log
(100000)
,
i.e., when
1
>
(5
a
)
a
(1
-
a
)
1
-
a
,
i.e., when
a
≤
0
.
42
.
iii) Now assume that all individuals are taxed
10000
and the proceeds are used to finance
vouchers that can be used to finance education. Do you expect some individuals to benefit from
such a policy?
Now net income is equal to
90000
,
but individuals receive a voucher worth
10000
.
6
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Individuals who attend public schools attain utility
U
V
P U
= (1
-
a
)
log
(90000) +
alog
(20000) =
log
(90000)
-
alog
(4
.
5)
and those who choose not to attend public schools attain utility of
U
V
P R
= (1
-
a
)
log
(
C
)+
alog
(
E
) = (1
-
a
)
log
((1
-
a
)100000)+
alog
(
a
100000) =
log
(1
-
a
)
1
-
a
+
loga
a
+
log
(100000)
,
provided that
E
=
a
100000
exceeds
20000
,
i.e., provided that
a >
0
.
2
.
Naturally if
U
V
P U
> U
V
P R
then such an individual will choose to attend a public school. This
happens when
log
(90000)
-
alog
(4
.
5)
> log
(1
-
a
)
1
-
a
+
loga
a
+
log
(100000)
,
which reduces to
1
>
10
9
(4
.
5
a
)
a
(1
-
a
)
(1
-
a
)
,
which happens when
a <
0
.
38
.
Note that we earlier required that consumption of education be at
least equal to
20000
,
i.e.,
a >
0
.
2
.
Therefore, we conclude that individuals
a >
0
.
38
go to private
schools.
Now, we can examine how welfare of a given individual changes when vouchers are introduced.
We need to consider a number of case.
Case 1
Initially chose public schools and continues at public school this happens for
a <
0
.
38
.
Utility
before was
U
P U
=
log
(100000)
-
alog
(5)
and utility with vouchers is
U
V
P U
=
log
(90000)
-
alog
(4
.
5)
.
Therefore, such agents lose when
log
(90000)
-
alog
(4
.
5)
< log
(100000)
-
alog
(5)
,
which reduces to
1
<
(
10
9
)
1
-
a
,
which holds for all
a
in this range. Hence those who initially went to public schools
lose when vouchers are introduced.
Case 2
Let us look at individuals who switched to private schools.
These are individuals for whom
a
∈
(0
.
38
,
0
.
42)
.
Initially they were at public and now they are at private. Their welfare initially
was
U
P U
=
log
(100000)
-
alog
(5)
and now it is
U
V
P R
=
log
(1
-
a
)
1
-
a
+
loga
a
+
log
(100000)
.
They
gain when
U
V
P R
> U
P U
,
which requires that
1
>
(5
a
)
a
(1
-
a
)
1
-
a
,
which implies that
a >
0
.
42
,
i.e.,
it does not happen in this range. Hence we can conclude that individuals in this range lose.
Case 3
These are individuals who were at private schools and remain in private schools. In this case
we have
U
V
P R
=
U
P R
and there is no change in welfare.
iv) Should such a policy involving vouchers be implemented if the government is interested in
increasing consumption of education by individuals who do not consume much education n the first
place?
Such a policy does not increase consumption of education of individuals who do not consume
much to start with. They remain at public schools and consume the same amount, but their welfare
goes down as they have to pay taxes to finance vouchers. It appears that such a policy should not
be implemented.
7
Problem #3
Consider an economy populated with
2
N
+1
individuals whose preferences are represented with
U
i
=
-
(
G
-
i
)
2
where
G
is the amount of public good provided.
i) What is the amount of
G
provided when decisions are made withe a simple majority vote?
Note that preferences are single peaked. Hence the optimal choice of the median agent,
i
=
N
+1
,
will be implemented, i.e.,
G
=
N
+ 1
.
ii) What is the amount of
G
provided when the government wants to maximized welfare given
with
W
=
-
(
G
-
1)
2
-
(
G
-
2)
2
-
(
G
-
3)
2
+
...
-
(
G
-
(2
N
+ 1))
2
?
In this case the optimal choice is given by
∂W
∂G
= 0
,
i.e.,
-
2(
G
-
1)
-
2(
G
-
2)
-
2(
G
-
3) +
...
-
2(
G
-
(2
N
+ 1)) = 0
,
which reduces to
(2
N
+ 1)
G
= 1 + 2 +
...
+ 2
N
+ 1 =
1
2
(2
N
+ 1)(2
N
+ 2) = (2
N
+ 1)(
N
+ 1)
,
i.e.,
G
=
N
+ 1
.
iii) Now, assume that the country allows for migration and the population by two individuals
whose preferences are given with
U
=
-
(
G
-
1)
2
.
What is the choice of
G
now if decisions are made
by voting?
In this case we allow two individuals in with low preferences for
G.
Now, there are
2
N
+ 3
individuals in total and the individual in the middle has index
i
=
N
-
1
.
Again, preferences are
single-peaked hence we can conclude that
G
=
N
-
1
will be chosen.
iv) What is the level of
G
chosen if the government decides to maximize the sum of all utilities?
As before the optimal choice is given with
∂W
∂G
= 0
,
which is equivalent to
-
2(
G
-
1)
-
2(
G
-
1)
-
2(
G
-
1)
-
2(
G
-
2)
-
2(
G
-
3) +
...
-
2(
G
-
(2
N
+ 1)) = 0
,
which reduces to
(2
N
+ 3)
G
= 1 + 1 + 1 + 2 + 3 +
...
+ 2
N
+ 1 = 2 +
1
2
(2
N
+ 1)(2
N
+ 2)
,
i.e.,
G
=
2 + (2
N
+ 1)(
N
+ 1)
2
N
+ 3
=
N
+
3
2
N
+ 3
.
8
Problem #4
Consider individuals with preferences represented with
U
(
C
) =
log
(
C
)
There are two possible
values for
C, C
L
and
C
H
.
Individuals are indexed with numbers from interval
[0
,
1]
.
Individual
a
∈
[0
,
1]
consumes
C
L
with probability
a
and
C
H
with probability
1
-
a.
i) What is the highest amount that agent
a
is willing to pay for complete insurance?
Without insurance an agent of type
a
attains expected utility of
EU
NI
a
=
alog
(
C
L
) + (1
-
a
)
log
(
C
H
) =
log
(
C
a
L
C
1
-
a
H
)
and with complete insurance that costs
z
she always ends up with consumption of
C
H
-
z
=
C
L
-
z
+
D,
where
D
=
C
H
-
C
L
is the value of the claim and her utility is
EU
W I
a
=
alog
(
C
L
-
z
+
D
) + (1
-
a
)
log
(
C
H
-
z
) =
log
(
C
H
-
z
)
.
She is willing to buy insurance as long as
EU
W I
a
≥
EU
NI
a
,
i.e., when
log
(
C
H
-
z
)
≥
log
(
C
a
L
C
1
-
a
H
)
,
which reduces to
z
≤
C
H
-
C
a
L
C
1
-
a
H
.
Therefore, an agent of type
a
is willing to pay up to
z
a
=
C
H
-
C
a
L
C
1
-
a
H
for insurance.
ii) Imagine that insurance companies charge a fixed fee of
x
per dollar of insurance. Identify
the fraction of agents that are willing to purchase insurance in this case.
Note that the value of the claim is
D
=
C
H
-
C
L
hence the dollar value of the premium is
xD
=
x
(
C
H
-
C
L
)
.
Hence, only individuals for whom
xD
=
x
(
C
H
-
C
L
)
≤
z
a
,
i.e.,
x
(
C
H
-
C
L
)
≤
C
H
-
C
a
L
C
1
-
a
H
,
i.e.,
x
(
C
H
C
L
-
1)
≤
C
H
C
L
-
(
C
H
C
L
)
1
-
a
,
i.e.,
1
-
a
≤
log
(
C
H
C
L
-
x
(
C
H
C
L
-
1))
log
(
C
H
C
L
)
,
i.e.,
1
-
log
(
C
H
C
L
-
x
(
C
H
C
L
-
1))
log
(
C
H
C
L
)
≤
a
will buy insurance.
9
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iii) How does the revenue collected by the insurance company change with
x
?
Assuming uniform distribution of types we can write
R
=
xD
(1
-
a
)
,
i.e.,
R
(
x
) =
x
(
C
H
-
C
L
)
log
(
C
H
C
L
-
x
(
C
H
C
L
-
1))
log
(
C
H
C
L
)
.
The revenue first increases with
x
and then it goes down with
x.
This is due to the fact that as
x
increases the problem of adverse selection becomes more severe. Note that
R
(0) =
R
(1) = 0
.
iv) Would you expect the revenue to increase if the insurance company could charge different
premia to different individuals?
If types are observable then insurance companies can simply charge
x
a
=
z
a
D
to individuals of
type
a
and then revenue is equal to
¯
R
=
ˆ
1
0
z
a
da
=
ˆ
1
0
C
H
-
C
a
L
C
1
-
a
H
da
=
C
H
-
C
H
ˆ
1
0
(
C
L
C
H
)
a
da
=
C
H
-
C
L
-
C
H
log
(
C
L
)
-
log
(
C
H
)
,
which exceeds the previous value as each agent pays the highest amount possible and there are no
adverse selection issues.
10
Problem #5
Assume that the price of consumption is
1
.
Imagine that we have two groups of people. The
first group has preferences represented with
U
=
C
+ 4
A,
where
C
denotes private consumption
and
A
denotes the dollar value of art consumed by the consumer. The preferences of the second
group are represented with
V
=
C.
Note that given our assumptions both utilities are expressed
in terms of money now. Imagine that there are
N
1
people in the first group and
N
2
in the second
group and that the government values the welfare of all people equally. Furthermore, assume that
the two groups are equally productive and each member of each group makes
Y
= 100
in income.
Finally, assume that the government can provide
$0
.
80
worth of art for each dollar spent.
i) Would you agree that the benefits of provision of art exceed the costs of provision
No. A dollar is converted into
$0
.
8
,
which is inefficient
ii) Would you agree that the welfare in the economy increases if all agent pay a tax of
1
to
finance art, but art must be consumer privately?
In this case revenue is equal to
(
N
1
+
N
2
)
*
1 =
N
1
+
N
2
and the total amount of art provided
is
A
= 0
.
8(
N
1
+
N
2
)
and individual consumption is equal to
A
N
1
=
0
.
8(
N
1
+
N
2
)
N
1
.
The welfare in the
economy is given with
W
=
N
1
W
1
+
N
2
W
2
=
N
1
(99 + 4
0
.
8(
N
1
+
N
2
)
N
1
) +
N
2
99 = 102
.
2(
N
1
+
N
2
)
.
The level of welfare increases in this case.
iii) Would you agree that the welfare in the economy increases if all agents pay a tax of
1
to
finance art and art is public good?
If art is public good then consumption can be done simultaneously and each individual consumes
A
= 0
.
8(
N
1
+
N
2
)
.
Hence, in this case welfare is given with
W
=
N
1
W
1
+
N
2
W
2
=
N
1
(99 + 4
*
0
.
8(
N
1
+
N
2
)) +
N
2
99 = (
N
1
+
N
2
)(99 + 3
.
2
N
1
)
and it increases with taxation as long as
N
1
is an integer.
iv) Would you answer to part iii) be different if the government was less efficient in providing
art with being able to deliver only
$0
.
50
worth of art for each dollar spent?
In this case we would have
W
=
N
1
W
1
+
N
2
W
2
=
N
1
(99 + 4
*
0
.
5(
N
1
+
N
2
)) +
N
2
99 = (
N
1
+
N
2
)(99 + 2
N
1
)
.
The answer would be the same as long as
N
1
is an integer.
11
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