ECON 210
MICROECONOMIC THEORY
FINAL EXAM, FALL 2005
PROFESSOR JOSEPH GUSE
Instructions. You have 3 hours to complete the exam. There are a total of 90
points available. It is designed to take about 1 minute per point. You are allowed to
reference a single page of notes, 2-sided. You may use a calculator, if you like, but
only to do simple calculations; you may not use any symbolic or graphing capabilities
which your calculator may have. There is space provided for each question. If you
need additional space, you may write on the back of the pages or use additional
sheets and staple them to your exam when you turn it in. Please show all of your
work. (Or at least enough so that the grader can figure out how you arrived at your
answers.) Please write your name on the exam itself.
Name:
Date: December, 2005.
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PROFESSOR JOSEPH GUSE
1. (a) (5 points) Suppose the Frank has exactly $12 to spend on beer and
pizza. Pizza is $1 per slice. Beer is priced at $2 per bottle for the first
4 bottles. Once someone has bought their first 4 bottles, they become
members of the Oktoberfest Club and then its just $0.50 per bottle. For
example, if you buy 3 bottles the total cost would be $6, but if you buy
6 bottles, the total cost would be $9, $8 for the first four plus $1 each
for bottles 5 and 6. Carefully draw a diagram showing Frank’s budget
set for combinations of beer and pizza. (You may assume that buying
fractions of a bottle is possible). Be sure to label intercepts as well as
marginal rates of transformation along each segment of the budget line.
Budget Set
12
Beer
(bootles)
MRT = 2
MRT =
4
1
2
4
12
Pizza
(Slices)
If Frank spent all of his money, $12, on pizza, he could buy 12
slices at $1 each as the horizontal intercept indicates. At first
giving a slice of pizza only yield a 12 bottle of beer since beer is
$2 per bottle, hence moving away from the horizontal intercept
along the budget, the MRT is equal to 21 . However once Frank has
bought 4 beers (and spend $8 on beer), additional beers are just
$0.50 each. Now giving up a slice of pizza frees up enough cash
to get 2 additional beers. Hence the MRT between (4, 4) and the
vertical intercept is 2. Notice that the vertical intercept is 12; If
Frank spent all of his money on beer, the first four would cost him
$8 leaving $4 which is enough for 8 additional beer at $0.50 each.
ECON 210
MICROECONOMIC THEORY
FINAL EXAM, FALL 2005
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(b) (5 points) Using one of the utility function types we have studied the
most (Perfect Substitutes, Perfect Complements, and Cobb-Douglas),
write down an exact specification of Franks utility which would be consistent with the observation that he demands exactly 4 beers. ANSWER.
Perfect Complements (a.k.a. Leontief) preferences would be consistent
with this observation.
(1)
u(z, b) = min {αz, αb}
Any choice of α will work (monotonic transformation), but it is critical
that the coefficients be equal. This is because the observation is that he
consumes these two good in equal amounts. Note that pefect substitutes
is not consistent with the observation. Any perfect substitute utility
specification would have Frank at a corner solution - either spending all
his money on beer or spending it all on pizza. Note also that CobbDougals is not possible...but this is the next problem.
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PROFESSOR JOSEPH GUSE
(c) (5 points) Suppose that Franks’s preferences are Cobb-Douglas:
u(b, z) = bα z (1−α)
Is it possible that for some choice of α between 0 and 1, Frank’s demand
for beer could be 4 bottles?. Explain why or why not. [Hint: In your
answer you may want to describe the relationship between MRS and
MRT to the right along the budget line and to the left along the budget
line.] ANSWER. No. This is not possible. Suppose that it were
possible. Then Franks’s MRS just to the left of (4, 4) would have to
be greater than 2 beers per slice AND his MRS just to the right of (4, 4)
would have to be less than 12 . In short his MRS would have to take a
discontinuous jump from 2 to 12 at (4, 4). But this is not consistent with
Cobb-Douglas preference whose indifference curves are smooth. Of our
standard 3 types of preferences, Leontief (perfect complements) is the
only one that would work. Indeed under the Leontief specification, MRS
does take a discontinuous jump from ∞ to 0 at (4, 4). Put another way,
it is impossible place a round object in a corner and have its rim touch
the very inner point of the corner.
ECON 210
MICROECONOMIC THEORY
FINAL EXAM, FALL 2005
5
2. (a) (5 points) Draw a diagram with today’s consumption on the horizontal
axis and tomorrow’s consumption on the vertical axis. Depict in this
diagram the budget set for someone who will earn $400 today and $1100
tomorrow who faces a one-period interest rate of .10. Be sure to carefully
label the intercepts.
(b) (5 points) In the same diagram show the preferences for some who would
borrow $200 today. Be sure to label their consumption today and tomorrow. [You may leave the numbers in expression form.]
Borrowing $200
Tomorrow
1100+(1.1)400
=1540
Endowment
1100
Consumption
Choice
1100-(1.1)200
=880
MRT=1.1
400 600
Today
1400=
400 + 1100
1.1
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PROFESSOR JOSEPH GUSE
(c) (10 points) Use a new diagram to explain the change on today’s consumtpion of an interest rate decrease from r′ to r for someone who would have
been a saver under the higher interest rate, r′ . In particular, your graph
should show the decomposition of the change in today’s consumption
into income and substitution effects. Assume that today’s consumptions
and tomorrow’s consumption are normal goods.
Rate Decrease Effect On Saver
Tomorrow
Original
Choice
SE
Compensated
Choice
IE
Old Level of
Utility(r’)
New
Choice
Compensated
BL
Old BL
MRT = (1+r’)
SE
IC representing
New Level of Utility (r)
New BL, MRT = (1+r)
Today
IE
Black Budget Line is Original one. It has steeper slope of (1 + r′ ).
Blue Line is new Budget Line. It reflects the new lower interest rate r and so has slope (1 + r). Green Line is the (Slutsky)
compensated Budget line. Note that the effect on tomorrow’s consumption is unambiguous since SE and IE act in same direction
toward less consumption tomorrow. However, effect on today’s
consumption may be ambiguous. As drawn, however, the substitution dominates and today’s consumption increases. (savings
decreases)
ECON 210
MICROECONOMIC THEORY
FINAL EXAM, FALL 2005
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3. Present Value
(a) (10 points) Your friend Mandy wants you to invest in her new bakery.
She promises that every dollar you invest will return $.40 per year for
3 years, starting in one year. If you don’t invest in her bakery you will
save your money in a savings account which pays interest at an annual
rate, r. Write down an expression involving r that tells you the present
value of a $1000 investment in Mandy’s bakery. Explain how you would
use it to make your decision of whether to invest the thousand or not.
ANSWER. Your $1000 investment would return $400 per year starting
in one year so must add toget the present value of each of those future
payments.
1000(.4) 1000(.4) 1000(.4)
+
+
(1 + r)
(1 + r)2
(1 + r)3
400
400
400
+
+
=
2
(1 + r) (1 + r)
(1 + r)3
PV =
Your investment is worthwhile if P V ≥ 1000.
(b) (5 points) If the interest rate is .05, what is value of an apartment which
costs $1000 per year to maintain and generates $11,000 per year in rental
revenue? [You may leave you answer in expression form for full credit.]
ANSWER.
11000 − 1000
.05
= 200, 000
PV =
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PROFESSOR JOSEPH GUSE
4. Avery is risk averse. [Note you should draw one diagram for this entire
problem]
(a) (5 points) Draw a picture of Avery’s utility over wealth outcomes. Put
wealth on the horizontal axis and utility on the vertical axis.
(b) (5 Points) There is a 50% of rain. Avery has $500 in a savings account.
He also owns a house. If it rains Avery’s house will flood and become
worthless. If it does not rain, Avery’s house is worth $1000. In your
picture mark the level of wealth Avery will have in both states of nature
as well as Avery’s expected wealth.
(c) (5 Points) Using the graphical method we discussed in class, label on the
vertical axis, Avery’s expected utility.
(d) (5 Points) Label the amount of certain wealth Avery would be willing to
take in exchange for both his house and his savings account.
(e) (5 Points) By identifying a segment of the horizontal axis, label the
minimum price Avery would ask for his house.
ECON 210
MICROECONOMIC THEORY
FINAL EXAM, FALL 2005
Risk Averse Avery
Eu
500 CE EW 1500
(1000)
(c) The expected utility is found by drawing a chord from the
point on the utility function representing the bad outcome to the
good outcome point, then draw a line up to that chord from EW,
the expected wealth. The Certainty Equivalent (d) is the amount
of the certain wealth Avery would take in exchange for his entire
portfolio. (e) The length of the red line (CE-500) is the least Avery
would be willing to accept in exchange for his house.
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PROFESSOR JOSEPH GUSE
5. John likes to go out in his canoe to get in a little fishing and enjoy the peace.
Ralph likes to go water-skiing in the same lake. Ralph’s willingness to pay
for the first hour of water-skiing per day is extremely high, but decreases
for each hour after that. In other words Ralph has decreasing MWTP for
waterskiing. Also assume that his MWTP does not reach zero until the 11th
hour of waterskiing.
Ralph doesn’t really mind John’s presence out on the lake. Having to steer
a course around his canoe, makes the water-skiing that much more fun. John,
on the other hand, often wonders just how quickly he would reach for his cell
phone if he noticed that Ralph were drowning. In fact, the more time Ralph
spends out on the lake, the more it bothers John and it bothers him at an
increasing rate. This increasing marginal damage function starts out very
low, but by the 11th hour of Ralph’s skiing, it is extremely high.
(a) (5 Points) Describe the efficient solution using a diagram with the number of hours Ralph spends waterskiing on the horizontal axis. ANSWER The efficient number of hours for Ralph to ski is the number of
hours which equates John’s marginal damage with Ralph’s MWTP.
(b) (5 Points) Describe how the efficient solution might be reached as an
equilibrium using a tax on water-skiing. ANSWER Set the tax equal
to the marginal damage evaluated at the efficient number of hours.
(c) (5 Points) Label the welfare effects of the tax (Ralph’s surplus changes,
John’s surplus changes, Government Revenue) compared to the case
where Ralph skis all that wants. ANSWER see diagram.
ECON 210
MICROECONOMIC THEORY
FINAL EXAM, FALL 2005
Efficient Water Skiing
Marginal
Damage
(to John)
$/hour
Marginal Benefit
(a.k.a MWTP)
(Ralph)
Ralph’s Loss
Government
Revenue
Optimal
Tax
0
John’s Gain
In Surplus Due
To Tax
Efficient
Hours
11
Ralph’s Skiing
(Hours)
Government revenue raised by the optimal tax is the green area.
Ralph loses consumer surplus equal to the green area and the red
area. In other words, Ralph’s surplus shrinks from the entire area
under his demand curve to just the area under his demand curve
above the tax.. John gains the area under his marginal damage
curve from 11 back to the efficient number of hours.
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