ECO 220 – Intermediate Microeconomics
Professor Mike Rizzo
First COLLECTED Problem Set – SOLUTIONS
This is an assignment that WILL be collected and graded. Please feel free to talk about the
assignment with your friends or with your group and I strongly encourage you to submit
your assignment as a group.
Assigned:
Due:
Monday, February 21st
Monday, February 28th
1. Nicholson, Chapter 2, Internet Exercise #1. I would really prefer it if you could draw
the budget constraints in EXCEL, but if not, please be sure to clearly draw your
handwritten ones. You might refer to this website for some examples of how to use
Excel for economics.
http://www.bized.ac.uk/stafsup/options/sheets/econ_index.htm. This is another site
you might explore with your free time http://www.wabash.edu/EconApp/About.html.
SOLUTION:
Weekly Budget Constraint
Price of X = $10, Price of Y = $20, Income = $52,000
Good Y
50
40.6
slope = -1/2
slope = -1/2
0
0
81.3
100
Good X
Marginal Tax Rates for Single Filers:
0-$7,150
10%
$7,151-$29,050
15%
$29,051-$70,350
25%
Compute your total taxes paid after the income tax. You pay 10% on your first $7,150 of
income, for a total of $715 for that bracket. You pay 15% on your NEXT (marginal)
$21,900 of income for a total of $3,285 for that bracket. You pay 25% on your NEXT
$22,950 of income, for a total of $5,738 in that bracket. Therefore, your total taxes for the
year are $715 + $3,285 + $5,738 = $9,737 for the year. (This means that your AVERAGE
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tax rate was 19.5%) We are plotting a weekly BC, so the loss in income due to taxes is
$187.26, leaving you with an after tax income of $812.74 per month. Since the relative
prices of the two goods have not changed, we see that the budget constraint will shift in by
the amount of the total taxes paid and the slope has not changed.
2. Nicholson, Chapter 2, Internet Exercise #2.
ANSWER: Gasoline was rationed in order to conserve RUBBER (it reduced the number of
miles driven and our rubber supply was cut off by Japan at the start of the war). Motorists
were restricted to 3 gallons of gas for nonessential purposes each week. When goods are
that are in short supply are rationed and prices are not allowed to rise to reflect scarcity
value, black markets will quickly arise – as in the case of ticket scalping. It is likely that
gasoline would sell in a secondary market at much higher prices. Many motorists would be
made worse off by scalping, as can be seen from them forced to locate at Pt. A below:
Indifference Curves and Budget Constraint
Gas Rationing and Black Markets
Other
Goods
A
B
Utility Level B
Utility Level A
0
3
6
9
12
Gasoline (gal / wk)
If gas were freely available at these prices, consumers would choose to be at point B,
consuming 5 gallons per week. These people would not only want to buy more gas at these
prices, but they would be willing to give up consumption of other goods to do this. Not all
motorists are made worse off by rationing, particularly those that wanted to consume 3 or
less gallons of gasoline per week. Presumably, this rationing has made them BETTER OFF,
as these folks will be able to turn around and sell their gas for prices far in excess of the
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going market rate. For more, read this article: http://www.cato.org/pubs/journal/cj15n14.html
3. On one occasion, Gowdy fills his car’s tank with gasoline on the evening before his
departure for a fishing trip. He awakens to discover that a thief has siphoned all but
1 gallon from his 21 gallon tank.
On another occasion, Gowdy plans to stop for gas on his way out the next morning
before he goes fishing. He awakens to discover that he has lost $20 from his wallet.
If gasoline sells for $1/gallon and the round-trip will consume 5 gallons, how, if at
all, should Gowdy’s decision about whether to take the fishing trip differ in the two
cases? (Assume that the inconvenience of having to refill the tank is negligible).
ANS: Suppose Gowdy’s income is $100 per month. Before his loss, he would be able to buy
100 gallons of gas per month or $100 worth of other goods. At the moment he discovers
his loss, his budget constraint will shift inwards in parallel fashion – he lost $20. So, now he
can consume $80 per month of gas or $80 of other goods. If he does not take the trip, he
will have $80 to spend on other goods in both scenarios. If he does take the trip, he will
have to purchase $5 worth of gas in either case. No matter what the source of the loss, the
remaining opportunities are EXACTLY the same. If his budget is tight, he may decide to
cancel his trip, otherwise he might go despite the loss. But, because the budget constraint
and tastes are the same whether he lost his cash or had his gas stolen, it would not be
rational for him to take the trip in one instance but not in the other.
4. Read the following paragraph from Steven Landsburg and then try and posit a
theory that suggests that depriving oneself of a future pleasure is CONSISTENT with
rational choice behavior.
First, the refrigerator locks. Why would any rational
creature want to erect an obstacle between itself and a
midnight snack? Midnight snacks have costs (usually measured
in calories or grams of fat), but they must also have
benefits--otherwise, they wouldn't tempt us. We snack when
we believe the benefits exceed the costs. In other words, we
snack when snacking is, on net and in our best judgment, a
good thing. What could be the point of making a good thing
more difficult?
But people do lock their refrigerators. They also destroy
their cigarettes, invest their savings in accounts that are
designed to discourage withdrawals, and adopt comically
elaborate schemes to force themselves to exercise. Odysseus
resisted the Sirens' call by lashing himself to the mast. I used to
have my secretary lock my computer in a drawer every
afternoon so I couldn't spend my entire day surfing the Net.
Economists have tried to explain such behavior in all sorts
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of unsatisfying ways. You can say that people like to avoid
making choices--but isn't the purchase of the lock a choice?
You can suppose that our minds house multiple "individuals"
with conflicting preferences--but it's unclear how to turn that
into a precise theory of exactly how many people we're sharing
our minds with, and how their conflicts get resolved. You can
throw up your hands and say that some behavior is rational and
some isn't, and this particular behavior is in the second
category--but that's tantamount to giving up without a fight. Or,
most unsatisfying of all, you can simply posit a "taste" for
self-control.
THEORY: Again, quoting from Landsburg:
The problem with that one is that once you allow yourself
to start positing "tastes" for everything under the sun, you
abandon all intellectual discipline--any behavior at all can be
"explained" by the assertion that somebody had a taste for it.
Economist Deirdre McCloskey warns against hollow triumphs
like, "Why did the man drink the motor oil? Because he had a
taste for drinking motor oil!" If you can explain everything,
you've explained nothing.
But in his entirely marvelous book How the Mind Works,
cognitive scientist Steven Pinker suggests that we can safely
posit a taste for self-control without opening the floodgates
that would allow us to posit a taste for drinking motor oil.
Here's why: Unlike a taste for drinking motor oil, a taste for
self-control confers a reproductive advantage.
When you snack at midnight, you get most of the benefits,
but your spouse (who cares about your health and
appearance) shares many of the costs. So a taste for locking
the refrigerator in the afternoon--even when you know that,
by a purely selfish calculation, you ought to make yourself a
giant hot fudge sundae every night--makes you more desirable
as a mate. Therefore, we shouldn't be surprised that natural
selection favored people with a taste for refrigerator locks.
What about people who aren't looking for mates or who
are already securely married? They have a taste for
self-control because their ancestors (who must have mated
successfully or they wouldn't have become ancestors) had
that taste. The bottom line is that it is intellectually honest to
explain behavior by positing surprising tastes, provided
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those tastes are useful in the mating game. Presumably the
sociobiologists and evolutionary psychologists have had this
idea all along, but economists have been slow to recognize
its significance.
Now as to the origin of the universe--or, as I prefer to
phrase the question, "Where did all this stuff come from?"--I
now believe that everything is made of pure mathematics. I
came to this insight from Frank J. Tipler's book The
Physics of Immortality, all of which is wonderfully
provocative and some of which is convincing. His point is to
take seriously the claims of those artificial intelligence
researchers who assert that consciousness can emerge from
sufficiently complex software. Pure mathematics is pure
software and contains patterns of arbitrary complexity. The
universe itself, together with the conscious beings who
inhabit it, could be one of those patterns.
Or maybe not. The argument only works if you believe that
mathematics is eternal and precedes the universe. One
could equally well argue that mathematics arises from counting
and measuring and so can't exist until after there is a universe
of things to count and measure. I should also say that while I
love the idea that the universe is nothing but a mathematical
model of itself, I've never met anyone else who found the idea
of "software without hardware" even remotely plausible.
But there might be a good economic reason why we're
stymied. Steven Pinker points out that understanding the origin
of the universe is not a terribly useful skill. It confers no
reproductive advantage, so there's no reason we should have
evolved brains capable of thinking about such a question.
Nature is too good an economist to invest in such frivolities.
On the other hand, the ability to understand human behavior
has clear payoffs for a social animal like Homo sapiens. So
it's not too much to hope that we could work out a detailed
and convincing theory of refrigerator locks.
5. Jones spends all of his income on two goods, X and Y. The prices he paid and the
quantities he consumed last year are as follows: PX = 10; X = 50; PY = 20; and Y =
25. This year, PX and PY are both 10, and his income is $750. Assuming his tastes
do not change, in which year was Jones better off, last year or this?
5
ANS: First, let’s draw his BC in year one. To do that, we need first to compute is income,
which is easy in this case. Income = PXX + PYY = (10 x 50) + (20 x 25) = $1,000. We can
now draw his budget constraint since we know the x-intercept will be 1000/10 = 100 and
the Y-intercept will be 1000/20 = 50. Represent his initial consumption bundle as point A.
After the change, the x-intercept is now 750/10 = 75 and the Y-intercept is 750/10 = 75
and you see that the budget constraint has rotated clockwise. We see that the new budget
constraint passes right through the initial consumption bundle at point A. Therefore, we can
say with certainty that Jones is NOT worse off than he was before. If his initial indifference
curves are flat enough (i.e. he likes good Y enough), this change has the possibility of
making him BETTER off.
Jones' Budget Constraint
Year 1: Price of X = $10, Price of Y = $20, Income = $1,000
Year 2: Price of X = $10, Price of Y = $10, Income = $750
Good Y
100
BC Year 2
75
50
A
25
BC Year 1
0
0
50
75
100
Good X
6. John buys shoes for $1 a pair and socks for $1 a pair. His annual income is $20
a. Draw John’s budget line.
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John's Sock and Shoe Purchases
Price of Shoes = $1/pr, Price of Socks = $1/pr, Income = $20
Socks (prs)
20
Initial Budget Line
0
0
20
Shoes (prs)
b. Now suppose that the government institutes two new programs: First, it
taxes shoes, so that shoes now cost John $2 a pair. Second, it gives John an
annual cash gift of $10. Draw his new budget line.
John's Sock and Shoe Purchases After Program
Price of Shoes = $2/pr, Price of Socks = $1/pr, Income = $30
Socks (prs)
30
Initial Budget Line
20
10
Initial Budget Line
0
0
10
15
20
Shoes (prs)
c. Suppose that with the new programs in place, John chooses to buy 10 pairs
of socks and 10 pairs of shoes. Has the government program made him
better off, worse off or neither?
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Just as with problem 5, we see that the program results in the new budget
constraint passing through the old budget constraint. After the change, we see
that John consumes as a bundle (10,10) which is on both budget constraints.
Clearly, this program has likely made John WORSE OFF. Why is this the case?
Let’s assume that John initially located at bundle (10,10). Unless his indifference
curves are extremely bowed in (i.e. look like L shapes), if he started at bundle
(10,10) this change would have induced him to increase his consumption of
socks and decrease consumption of shoes so that the new bundle lie up and to
left of initial bundle. He therefore would have been able to achieve a higher
level of satisfaction. Since he still consumes 10 shoes after the tax on shoes tells
us he gets great satisfaction from shoes and less so from socks, so his
indifference curves are likely to be fairly steep at this point. Suppose he initially
purchased less than 10 pairs of shoes – it must be the case that his indifference
curve passed ABOVE the bundle (10,10), otherwise he would have chose to
locate at bundle (10,10). Since the new budget constraint lies ABOVE the old
budget constraint when shoe purchases are less than 10, it must mean that the
new indifference curve reached after the government program would still pass
above the initial (10,10) bundle. Therefore, this initial situation is impossible.
Now, consider the possibility that John consumed more than 10 pairs of shoes
before the program. It must also be the case that his initial indifference curve
passed ABOVE the bundle (10,10), otherwise he would have chosen it.
Therefore, since we know that after the change he locates at (10,10) he must be
WORSE off than he was before the change.
7. Herman has an income of $10, which he spends on fishheads and all other goods.
Fishheads cost $1 apiece.
a. Suppose that the government agrees to pay half of Herman’s fishhead bill, so
that fishheads now cost him only $0.50. He now chooses to buy 8 fishheads.
Show how the government program affects Herman’s budget line, and show
his new optimum point. Call it P. What are the coordinates of the point P?
(8 fish heads, $6.00)
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Herman's Fish Head Purchases Post Program
Price of Fish Heads = $0.50 each, Income = $10
Income ($)
10
P
6
0
20
8
Fish Heads
b. Now suppose the government ends the program in part (a) and replaces it
with a new and simpler program: Herman just gets a cash gift of $4. Show
his new budget line. Does it go above, below or through point P? How do
you know?
Herman's Fish Head Purchases Income Program
Price of Fish Heads = $1.00 each, Income = $10 + $4 gift
Income ($)
10
P
6
0
8
14
20
Fish Heads
Since Herman is consuming 8 heads and $6 of other goods, at original price of fish this
would cost $14. This is exactly how much income Herman would have under the transfer
program.
c. Of the two programs in parts (a) and (b), which is more expensive for the
government? Which does Herman prefer? Justify your answer.
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In part (a), the government pays Herman $0.50 for every single fish he buys. Since he
chooses to buy 8 fish, the program must cost the government $4.00. In part (b), we know
the government makes a cash transfer of $4. Therefore, the two programs cost the
government the SAME.
Herman's Fish Head Purchases Program Comparison
Price of Fish Heads = $1.00 each, Income = $10 + $4 gift
Income ($)
10
Cash Transfer
P
6
I nitial
Subsidy
0
8
14
20
Fish Heads
Intuitively, it should be clear to you that Herman prefers a cash transfer to a subsidy, but
why must this be the case in this example? Had he preferred the subsidy, in that scenario
he would have selected a bundle that was well to the Southeast of point P. So, we know
that bundle P is preferred to any other bundle along the “subsidy” line to the Southeast of
P. Since all points on the “Cash Transfer” budget constraint that are to the Northwest of P
lie ABOVE all points on the “Subsidy” budget constraint to the Northwest of P, it must be the
case that the Cash Transfer is preferred. If you are having trouble convincing yourself of
this fact, note that if the picture began at Point P instead of going all the way down to the
x-axis where income = $0, what would indifference curves have to look like for Herman to
choose a bundle at Point P and not one to the Northwest of it? It would have to be nearly
vertical, indicating that he has a strong taste for fish heads. However, we just showed
above that his preferences do not exhibit this shape for if they did, he would have located
to the Southeast of P during the “Subsidy” program.
8. Suppose that the quantity of education, measured by classroom-hours per year is
fixed and that when we speak of spending more money on education, what we
mean is not buying more hours of education, but rather buying education of higher
quality. Suppose that the current tax system charges each family a tax of PE for 1
unit of education at the quality of education currently offered in public schools.
Furthermore, everyone is taxed this amount, whether or not they use the public
school. If a family chooses not to send its child to public school, it can purchase 1 or
10
more units of education at a private school, also at a price of PE. For example, to
purchase 1.5 units of education at a private school would mean purchasing an
education that is of 50% higher quality than that currently offered in a public school.
Furthermore, families are required by law to provide their child with at least one unit
of education, public or private.
a. Draw the Smith family’s budget constraint in the absence of the education
tax program. Assume the family income is $10,000 per year and the price of
education, PE is $2,500 per unit.
Smith Family Schooling, No Taxes
Price of School Quality = $2,500/unit , Income = $10,000
Income ($)
$10,000
0
1
2
3
4
Educational Quality
b. Draw the Smith family’s budget constraint AFTER the tax program. It would
be useful to note that the 1st year of public education is free under this
program. It will also help you to remember that to attend a private school
means spending an additional $2,500 per unit of education, and withdrawing
from the public school.
To draw this, ask yourself the question, “even if Smith kid dropped out of public school and
consumed zero units of school quality, how much money would the family have left?” The
answer is $7,500 – because everyone pays the tax. Since the entire first quality unit is free
if the student attends public school, the budget constraint should have a slope of ZERO
between 0 and 1 (recall that budget constraint equals the (negative) relative price of the
good under consideration.
Now, the family has a choice to make, even if they only want to buy one unit of educational
quality, they can do it either in the private school OR public school. If in the public school,
the family will still have $7,500 in their pockets after one unit is purchased. However, if the
family attends private school, they DO NOT get a refund on their tax payment. In other
words, to purchase one unit of quality education at a private school, the family must still
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pay its taxes AND THEN also pay an additional $2,500 in tuition for that first unit of school
quality at the private. Therefore, for a family that purchases one unit of private school
quality, they will only have $5,000 in their pockets. If families want to purchase more than
1 unit of school quality, they must do it at the privates, since we assumed the quality of
publics is fixed. Therefore, beyond 1 unit of school quality, the slope of the budget
constraint MUST be equal to the (negative) price of private schooling.
Income ($)
Smith Family Schooling, Tax Program
Price of School Quality = $2,500/unit , Income = $10,000
One unit mandatory school, $2,500 tax
$10,000
One unit of public quality purchased
$7,500
$5,000
One unit of priv ate
quality purchased
0
1
2
3
4
Educational Quality
c. Where will families with regular shaped indifference curves likely locate? i.e.
how much education are they likely to purchase? What about families with
steeply sloped indifference curves?
Most families will locate at the “kink.” In other words, unless indifference curves are
extremely steep (indicating a huge preference for educational quality), families will choose
to send their children to public schools. (Try to draw an I.C. that is tangent to the sloped
portion of the BLUE budget constraint, but that rises ABOVE the kink …)
d. Repeat parts (a) and (b) for the Jones family whose income is $50,000 per
year.
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Jones Family Schooling, No Taxes
Price of School Quality = $2,500/unit , Income = $50,000
Income ($)
$50,000
0
Income ($)
5
10
15
20
Educational Quality
Jones Family Schooling, Tax Program
Price of School Quality = $2,500/unit , Income = $50,000
One unit mandatory school, $2,500 tax
$50,000
One unit of public quality purchased
$47,500
$45,000
One unit of priv ate
quality purchased
0
5
10
15
20
Educational Quality
e. Suppose that the Smith family accepts the government’s offer of free
education and the Jones family rejects it, is it still possible that each family
has the SAME tastes for education? What does this tell you about why
people of different income backgrounds tend to locate in different
educational sectors?
Yes, it is entirely possible for each family to have the same tastes for education. For poorer
families, educational expenditures make up a large share of their family budgets, therefore
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even if they care about good schooling as much as their wealthier counterparts, they simply
cannot afford to purchase the quantity of quality schooling that their counterparts do.
f.
In recent years there has been much debate about the need to improve the
quality of elementary and secondary education in the U.S. One popular
method has been to increase school choice for families. By providing families
with a voucher that could be used toward tuition at any school of the family’s
choosing, it would expand the options available to poor families and
introduce competition in the market for educational services.
i. Under a voucher system, families all still are forced to pay PE in school
taxes. Each family would then receive a voucher in the amount
equivalent to PE that may be used toward the purchase of public or
private education. The law still requires families to provide at least
one unit of education. Draw the B.C. for the Smith family after the
implementation of the voucher program.
Income ($)
Smith Family Schooling, Voucher Program
Price of School Quality = $2,500/unit , Income = $10,000
One unit mandatory school, $2,500 tax, $2,500 voucher
$10,000
Old B.C.
One unit of public quality purchased
$7,500
$5,000
One unit of priv ate
New B.C.
quality purchased
0
1
2
3
4
Educational Quality
ii. How much schooling is the Smith family likely to purchase after this
program is implemented?
The Smith family is much more likely to purchase MORE than 1 unit of school quality after
this program is implemented. The voucher program eliminates the discontinuity in the
budget constraint as families no longer have to forfeit their taxes if they do not choose to
attend a public school.
9. Suppose that you are a government policymaker and your goal is to increase the
well-being of poor people. You can do so by subsidizing their food, education, and
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medical care; or you can do so by giving them cash.
a. On the basis of your answer to #7 above, make an argument in favor of
giving cash.
Giving cash will increase the utility of poor people beyond giving a subsidy for consumption.
This is because the cash grant will allow people to purchase the goods that provide them
with the most satisfaction. Furthermore, a subsidy can often be treated as a cash grant
(think of the food stamp program) by recipients because these subsidies allow families to
transfer more of their spending to other items.
b. Now suppose that although you want to help only the poor, it is difficult for
you to tell who is poor and who is rich, and you are worried that some rich
people will claim a share of the cash giveaways. Can you make an
argument for subsidizing education instead of giving away cash?
This may be a mechanism by which one can determine who is poor and who is rich by the
rich families self-selecting out of the public schooling sector (see #8 above). Therefore,
though the utility of the poor would not be as high as under a cash grant scheme, the
“leakage” in the program is likely to be much smaller. Leakage refers to the percentage of
program benefits that go to non-targeted populations.
c. Can you think of a reason why governments would want to deliberately limit
the amount of education available at public schools?
If too much education is available at public schools, then they will be chosen by both poor
and rich students. If the purpose of public schooling is to subsidize the poor at the expense
of the rich, expanding the public system would tend to defeat that purpose.
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