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Unformatted text preview: Econ 100A – Microeconomics
Professor Reynolds
Department of Economics
University of California, Berkeley Problem Set 1 Solutions
1. Question 2.11 part A in the book & parts C & D & E below. For part A, the graphs do
not have to be done separately as long as they are well labeled.
ANSWER A. a, b, c: The graph shows two lines representing the budget sets for a consumer choosing between
Other Consumption and Miles Driven (dashed) and Other Consumption and Miles Flown
(solid), respectively. We have normalized the per unit price of Other Consumption to $1
so that if a consumer spends his budget on Other Consumption he may consume 10,000
units. Note that the budget line associated with Miles Flown is kinked, because after
traveling 25,000 miles the unit price of flying additional miles is halved.
To see how frequent flyer perks could induce consumers to fly more than they otherwise
would, consider what the budget line for flying would look like if the price did not drop
after 25,000 miles. In this case the per unit price of driving would be less than the per unit
price of flying for all quantities of travel. The most preferred indifference curve that a
consumer could reach under the budget set would always be a bundle consisting of a combination of Miles Driven and Other Consumption. Introducing frequent flyer perks,
as in the graph above, creates a section of the budget set where the consumer could
consumer more Other Consumption and Miles Flown, possibly reaching a higher
indifference curve.
C. Now graph the budget constraint in terms of miles only, with miles flown on the y axis
and miles driven on the x axis. Write the mathematical form of the budget constraint.
ANSWER Budget Constraint
Note that because relative prices change when the consumer chooses to fly more than
25,000 miles, the budget constraint will consist of two line segments. For bundles
including 25,000 miles flown or fewer, the constraint is immediate:
10000≥0.16x1+0.2x2
For bundles including more than 25,000 miles flown we must incorporate the fact that the
consumer must pay one price for the first 25,000 miles flown and a second price for any
additional miles flown.
10000≥0.16x1+0.2(25000)+0.1(x2−25000) 7500≥0.6x1+0.1x2
Incorporating the nonnegativity constraints the budget set is described as follows:
10000≥0.16x1+0.2x2:Ifx2≤25,000
7500≥0.6x1+0.1x2:Ifx2>25000
x1≥0
x2≥0
D. Suppose the airline changes its policy so that instead of giving you a price discount,
frequent flier miles accrue yearly. Now you have $10,000 in your budget and an
additional 5,000 miles that you earned last year. How do your figures in parts A.a & C
change? Draw new graphs.
ANSWER In each case the budget set is expanded by the additional availability of frequent flyer
miles. However, the frequent flyer miles are nontradable so the intercepts for Other
Consumption and Miles Driven do not change. The dashed lines in the graphs below
capture this effect.
E. Now suppose that the 5,000 frequent flier miles are transferable and your friend is
willing to buy your miles at $0.15 per mile. Again indicate the changes in parts A & C.
You may indicate in the graphs drawn in part D.
ANSWER
Again the budget set is expanded by the additional availability of frequent flyer miles.
However, now the frequent flyer miles are tradable so the intercepts for Other
Consumption and Miles Driven do change. One may think of the opportunity to sell
frequent flyer miles as adding $750 to the initial endowment. The dashed lines in the
graphs below capture this effect. 2. Question 3.5 parts A & part C below
ANSWER
A a. To determine the greatest possible consumption for the current period we must
calculate the net present value of each of the future payments:
x(1.1)=100,000→x=90,909.09
y(1.1)5 =100,000→y=62,092.13
z(1.1)10=100,000→z=38,554.33
x+y+z=191,555.55
A b. To determine the greatest possible consumption in 10 years we must calculate the
future values of each of the stream of payments:
x(1.1)9=100,000→x=235,794.77
y(1.1)5=100,000→y=161,051
z=100,000
x+y+z=496,845.77
C. Suppose at age 29 you’ve spent all thus far so you are only awaiting the 100,000 in 1
year. However, the labor party gets elected and a 25% tax is imposed on unearned
income. You can get around this tax by a deal with your trust manager. She will “pay”
you $1,000 per week to chauffer her kids to school. Any money left “unearned” is
subject to the tax. What is the budget constraint for consumption in year 10 vs. bachelor
party spending? (Assume 52 weeks in a year and include weeks of work as a variable,
along with consumption in period 10 & bachelor party spending.)
ANSWER
Here we assume that any consumption in period 10 is financed by borrowing funds at the
beginning of the year against the payment on your 30th birthday, so that 10% interest will
accrue to any funds borrowed for consumption in period 10. Letting w represent work, x1
represent consumption in period 10 and x2 represent bachelor party spending we can
represent the budget line as:
0.75(100000−1000w)+1000w≥1.1x1+x2
75000+250w≥1.1x1+x2
Adding the nonnegativity constraints and recognizing that you can work at most 52
weeks per year, we arrive at the constraints for the budget set:
75000+250w≥1.1x1+x2
0≤w≤52
0≤x1
0≤x2 3. Question 3.8 parts A & B.a (not part B.b. though you may wish to do part B.b. on your
own as it will help you check that you got the right answer). For part A.d. do not give
both an intuitive & a graphical answer – explain your answer using the graphs only.
ANSWER
A a. and A b. The graph below contains the two budget lines for the higher (solid) and
lower (dashed) borrowing rates. A c. The set described in part a. is convex and the set described in part b. is not.
A d. I would prefer the scenario with the lower borrowing and higher saving interest
rates. As you can see from the above graph, if I prefer any amount of saving or
borrowing, the lower borrowing and higher saving rate scenario allows be to consume
bundles that include strictly greater quantities of consumption this summer and next
summer.
B a. The key here is to recognize that the budget constraint will be a piecewise function
that depends on whether the consumer is saving or borrowing. If the consumer is A c. The set described in part a. is convex and the set described in part b. is not.
A d. I would prefer the scenario with the lower borrowing and higher saving interest
rates. As you can see from the above graph, if I prefer any amount of saving or
borrowing, the lower borrowing and higher saving rate scenario allows be to consume
bundles that include strictly greater quantities of consumption this summer and next
summer. borrowing, the sum of the endowments must be greater than or equal to the sum of
consumptionkey here is to recognize that the budget constraint will be a then the sum of the
B a. The and the interest payment. If the consumer is saving piecewise function
endowments and the interest earned onis saving or borrowing. If the consumer is to the sum
that depends on whether the consumer savings must be greater than or equal
borrowing, the
of consumption. sum of the endowments must be greater than or equal to the sum of
consumption and the interest payment. If the consumer is saving then the sum of the
e1+e2endowments1and1)+c2interest earned on savings must be greater than or equal to the sum
≥c1+(rB)(c −e the :Ifc1≥e1
of consumption.
e1+(rS)(e1−c1)+e2 ≥c1+c2:Ifc1<e+ e " c + (r )(c # e ) + c : If c " e
e1 1 2 1
B
1
1
2
1
1
c1≥0
e1 + ( rS )(e1 # c1 ) + e2 " c1 + c 2 : If c1 < e1
c2 ≥0
c1 " 0
c2 " 0 4. Question 4.4 part A 4. Question 4.4 part A ANSWER
A a. ANSWER ! A a. A b. Yes. If A is on a higher indifference curve than C and preferences are strictly
convex, then A and C must be on a higher indifference curve than B. That B is chosen
over C on Tuesday implies that the consumer’s preferences have changed.
A c. Yes. You can conclude that the indifference curve containing bundles A, B and C is
linear between A and B. To see this note that for A and B to be weakly preferred to C
they must be on at least the same or higher indifference curves. However, if they were on higher indifference curves the consumer’s preferences would not be weakly convex.
Thus, A, B and C must be on the same indifference curve, so the indifference curve is
linear between bundles A and B and passes through bundle C.
5. Question 4.5 parts A.a. & A.d. & B.c.
ANSWER
A a. Yes. Trading 1.5 Apples for a Banana would make both consumers better off. To See this
note that at the current rate of marginal substitution, the consumer with 3 bananas and 6
apples would be indifferent between his current bundle and one in which he gets 4 apples
and 4 bananas. Similarly, the consumer with 10 apples and 5 bananas would be
indifferent between his current bundle and 11 apples and 4 bananas. If the two consumers
trade 1.5 apples for a banana then the first consumer will have 4.5 apples and 4 bananas
and the second consumer will have 11.5 apples and 4 bananas. Because consumers are approximately locally linear then, at the current rate of marginal substitution, the
consumer with 3 bananas and 6 apples would be indifferent between his current bundle
and one in which he gets 4 apples and 4 bananas. Similarly, the consumer with 10 apples
and 5 bananas would be indifferent between his current bundle and 11 apples and 4
bananas. If the two consumers trade 1.5 apples for a banana then the first consumer will
save 4.5 apples and o less, each of these bundles should be preferred to the current
htrictly prefer more t4 bananas and the second consumer will have 11.5 apples and 4
endowment.
bananas. Because consumers strictly prefer more to less each of these bundles should be
preferred to the current endowment.
A d. The marginal rates of substitution would have to be the same. If the marginal rates
o substitution were the same, then any trade have to be the same. If the both consumer
Af d. The marginal rates of substitution wouldthat would be acceptable to marginal rates s
could only leave them equally as well off as that would be acceptable
of substitution were the same, then any trade the current endowment. to both consumers
could only leave them equally as well off as the current endowment.
B c. To determine which utility functions represent the same tastes we must calculate and
c c. To determine which utility functions represent the same tastes we must calculate
Bompare the marginal rates of substitution. If the marginal rates of substitution are theand
compare tutility functions represent the same underlying preferences.
same the he marginal rates of substitution. If the marginal rates of substitution are the
same the utility functions represent the same underlying preferences.
#x
MRS1 = " 2
$x1
#
MRS2 = "
$
#x
MRS3 = " 2
$
#x
MRS4 = " 2
$x1
#x
MRS5 = " 2
$x1
$
(1 " # ) x 2 +1
#x1$ +1
MRS7 = DNE
#x
MRS8 = " 2
$x1 MRS6 = " The first, fourth, fifth and eighth utility functions represent the same underlying
preferences.
The first, fourth, fifth and eighth utility functions represent the same underlying
!
preferences. ...
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 Spring '08
 Woroch
 Microeconomics

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