Assignment 2
ECON3122, Intermediate Microeconomics
Due: Thursday, February 7th , by end of class
Directions: Answer all questions completely. Note the due date of the assignment. Late
assignments will be accepted at the cost of 15 points per day, up until 3pm on Tuesday, February
12th . At that time I will return the graded assignments and post the answers on the web. You
may turn in assignments to me after that time so that I can check your work for you, but please
realize that you will not receive a grade for the assignment. You may work in a group consisting
of up to 3 members –for each group please turn in only 1 set of answers and make sure all group
member names are on that set of answers. All group members will receive the same grade.
1
Consumer Choice (55 points)
Suppose that a consumer has the following utility function:
U (QA ; QB ) = QA + QB
This is a utility function for perfect substitutes. Assume that Y = $50 and that PA = $2 and
PB = $5.
1. (5 points) Calculate the utility of the following bundles:
(QA = 6; QB = 12)
(QA = 18; QB = 0)
(QA = 8; QB = 8)
(QA = 0; QB = 18)
Answer:
(QA = 6; QB = 12) : U = 18
(QA = 18; QB = 0) : U = 18
(QA = 8; QB = 8) : U = 16
(QA = 0; QB = 18) : U = 18
2. (5 points) Write down the consumer’s budget constraint, using the amounts given in the set-up
of this question to …ll in for any exogenous variables in the budget constraint.
Answer:
The budget constraint is:
1
Y = PA QA + PB QB
Plugging in the numbers we get:
50 = 2QA + 5QB
3. (5 points) How many units of good A would the consumer purchase if he purchased 0 units
of good B? How many units of good B would the consumer purchase if he purchased 0 units
of good A?
Answer:
The consumer would purchase 25 units of good A if he purchased 0 units of good B.
The consumer would purchase 10 units of good B if he purchased 0 units of good A.
4. (10 points) What is the Marginal Rate of Transformation (MRT) of goods A and B? What is
the Marginal Rate of Substitution (MRS) of goods A and B? (Hint: The goods are perfect
substitutes.)
Answer:
M RT = 52 if good A is on the y-axis; M RT =
budget constraint and solve for QA to see this:
50
=
QA
=
2
5
if good B is on the y-axis. Just take the
2QA + 5QB
5
QB
25
2
M RS = 1 regardless of which axis the goods are on – these goods are perfect substitutes,
so the indi¤erence curves are straight lines with slope of ( 1).
5. (10 points) Draw a graph (not necessarily to scale, but representative of the problem) that
includes at least 2 indi¤erence curves and the budget constraint. Be sure to label the axes
of your graph.
Answer:
2
Qty. A
30
20
10
0
0
5
10
15
20
25
30
Qty. B
The red line is the budget constraint adnd the green lines represent indi¤erence curves of 25,
18, and 10. The key is that the budget constraint has a steeper slope than the indi¤erence
curves IF good A is on the y-axis. If good B is on the y-axis, then the budget constraint will
have a ‡atter slope than the indi¤erence curves.
6. (10 points) Find the consumer’s optimal bundle. What is his utility at this bundle? (Hint:
Use your graph in part 4 as an aid.)
Answer:
The optimal bundle for this consumer is 25 units of good A and 0 units of good B. Looking
at the graph above, we see that the consumer can reach the highest indi¤erence curve if he
simply consumes all of good A. Intuitively, you have 2 goods that are perfect substitutes,
and one of the goods costs less than the other. Why would you ever buy the more expensive
good?
7. (10 points) At the optimal bundle, is the condition that MPUAA = MPUBB satis…ed? Explain why
or why not. (Again, if the graph is drawn correctly it may help.)
Answer:
The condition is not satis…ed because we are at a corner solution. We also know that the
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slope of the indi¤erence curves are ( 1) and the slope of the budget constraint is either
2
2
or
5 depending on how you drew the graph. Since both the M RS and the M RT are
constants (and are not exactly equal) we will not have MPUAA = MPUBB .
2
Indi¤erence Curves (36 points)
Look at the following indi¤erence curves. Explain why the indi¤erence curves in each graph violates
(at least) one of the properties of consumer preferences. Note that the 2nd indi¤erence curve is 2
3
bundles thick. (4 points each)
1.
2.
3.
Answer:
NOTE:
The …rst indi¤erence curve violates the more is better principle. We can show that there is
a point (call it point A) on the indi¤erence curve such that point A has more of both goods at a
di¤erent point on the lower portion of the indi¤erence curve.
4
The second indi¤erence curve also violates the more is better principle. If the indi¤erence curve
is 2 or more bundles thick, then there is a bundle on the indi¤erence curve that is NE of a bundle
on the same indi¤erence curve. This is a clear violation of more is better.
The third indi¤erence curve violates more is better and transitivity. Let E represent where
the two indi¤erence curves cross. Let A be a bundle on I2 that is to the right of point E. Let B
be a bundle on I1 such that the quantity of good 2 consumed lies between the quantity of good 2
consumed at E and the quantity of good 2 consumed at B. The indi¤erence curves show us that
this consumer is indi¤erent between bundles E and A and is also indi¤erent between bundles E and
B. So the consumer must be indi¤erent between the bundles A and B. However, by construction
B has a larger quantity of both good 1 and good 2 than A, so A must be preferred to B.
On a separate sheet of paper, draw at least 3 indi¤erence curves for each set of preferences listed
below. Label the lowest indi¤erence curve I1 , the middle indi¤erence curve I2 , and the highest
indi¤erence curve I3 . (8 points each)
1. Put left shoes on the y-axis and right shoes on the x-axis. Russ must have a left shoe to go
with each right shoe.
Answer:
left shoes
4
3
2
1
0
0
1
2
3
4
5
right shoes
2. Put sardines on the y-axis and pizzas on the x-axis. Tim gets zero utility from consuming
pizzas but enjoys sardines immensely.
Answer:
5
Sardines
5
4
3
2
1
0
0
1
2
3
4
5
Pizza
3. Put Georgia-Paci…c copier paper on the y-axis and Xerox copier paper on the x-axis. Karen
is indi¤erent between which type of paper she uses.
Answer:
Georgia-Pacific
3
2
1
0
0
1
2
3
4
5
Xerox
The lowest indi¤erence curve is the line with the y-intercept at 1, the middle indi¤erence
curve is the line with the y-intercept at 2, and the highest indi¤erence curve is the line with
the y-intercept at 3. Note that all 3 of these indi¤erence curves have a slope equal to ( 1).
3
One good shoe (9 points)
If you have ever gotten a ‡at tire on your car and the tire is beyond repair you know that you can
get a single tire replaced on your car and that a market exists for this good (the one tire you need)
and this service (changing the tire). But what about shoes? Say that something happens to one
of your shoes – your dog, instead of eating your homework, decides to eat just your right shoe of
your brand new pair of shoes. Is there a similar market for one good shoe? In other words, can
you …nd just a right shoe for sale somewhere or do you need to purchase a new pair of shoes? If
you know of such a market provide some evidence. If you believe that no market for “one good
shoe”exists, explain why a market for “one good shoe”does not exist while a market for “one good
tire” does exist.
6
Answer:
Apparently there is a market for just one shoe at www.mixmatchshoes.com. I was unaware of
that retailer and I could not …nd one that sold just one shoe when I searched. All the domain
names “www.onegoodshoe.(com, biz, org)” were unregistered when I looked them up, so you can
start a rival to www.mixmatchshoes.com if you like using one of those names. But you should know
that the mixmatchshoes.com site only went into operation on November 12, 2007, so it is only a few
months old. Thus, there really wasn’t much of a market for one good shoe prior to that, although
someone in class noted that when growing up it is fashionable to wear two di¤erent Chuck Taylor’s,
and so a market for one shoe did exist (I am uncertain from the answer as to whether one could
buy one Chuck Taylor in the store, or mix and match in the store, or if trades occurred between
two kids on the block with like sized feet). And if (insert today’s pop culture icon) were to start
wearing two di¤erent branded shoes (a left Nike and a right Adidas) then my guess is many people
would follow this person’s lead. As to why the market is not that large (only one retailer has been
found), there are many reasons. Not enough demand for one shoe is probably the primary one.
For manufacturers of shoes, it probably doesn’t make sense to change their operations to be able
to produce only one of a pair of shoes. For middlemen (like mixmatchshoes.com) it is probably
likely that the transactions costs of acquiring the pairs and selling them individually is quite high,
particularly prior to the rise of e-commerce. The tra¢ c through a traditional brick and mortar
store would probably not be enough to make the store pro…table.
As for di¤erentiating between tires and shoes, some people suggested that functionality of the
shoes was di¤erent than that of tires. In essence, any tire can replace any tire at any location
on the car, whereas only left shoes can replace left shoes and right shoes can replace right shoes.
Some suggested that brand didn’t matter for functionality of tires, whereas brand did matter for
functionality of shoes (this was disproved in class on 2/12). Others suggested that wearing two
di¤erent brands of shoes would make one "unfashionable", which I suppose is possible, and people
don’t want to be unfashionable. Some suggested that the cost of tires was so much greater than
that of shoes that people were willing to buy just one tire to save on cost while they would not be
willing to buy just one shoe. However, I found a pair of Nike Air Max ’95 360 Men’s shoes cost
$200, while I can buy 4 tires (without installation) from Tire Kingdom for a 2001 Ford Escort for
$49.99 apiece.
4
Bonus (5 points)
Suppose we have a demand function that is:
ln (QD ) = a
b ln (Pown ) .
Show that the coe¢ cient on ln (Pown ) is the own-price elasticity of demand. Note: You will probably
need to use some calculus to do this.
Answer:
We can di¤erentiate both sides with respect to Pown :
d ln (QD )
dPown
d ln (QD )
dPown
d ln (Pown )
dPown
1
b
Pown
=
b
=
7
Now we can use the chain rule to …nd:
d ln (QD )
d ln (QD ) dQD
=
dPown
dQD dPown
so that
d ln (QD ) dQD
dQD dPown
1 dQD
QD dPown
Pown dQD
QD dPown
Note that
dQD
dPown
Pown
QD
=
=
=
b
1
Pown
1
b
Pown
b
is just the general formula for own-price elasticity of demand.
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