Intermediate Microeconomics
Problem Set 2
Spring 2013
1. Shirley thinks that two 8-ounce cans of beer are exactly as good as one 16-ounce
can of beer. Suppose that these are the only sizes of beer available to her and that
she has $30 to spend on beer. Suppose that an 8-ounce beer costs $0.75 and a
16-ounce beer costs $1.
(a) At these prices, which size can will she buy, or will she buy some of each?
(b) Suppose that the price of 16-ounce beers remains $1 and the price of 8-ounce
beers falls to $0.55. Will she buy more 8-ounce beers?
(c) What if the price of 8-ounce beers falls to $0.40? How many 8-ounce beers
will she buy then?
(d) If the price of 16-ounce beers is $1 each and if Shirley chooses some 8-ounce
beers and some 16-ounce beers, what must be the price of 8-ounce beers?
2. Mary’s utility function is
,
100
, where
is the number of
silver bells in her garden and is the number of cockle shells. She has 500
square feet in her garden to allocate between silver bells and cockle shells. Silver
bells each take up 1 square foot and cockle shells each take up 4 square feet. She
gets both kinds of seeds for free.
(a) To maximize her utility, given the size of her garden, how many silver bells
and cockle shells should Mary plant?
(b) If she suddenly acquires an extra 100 square feet for her garden, how much
should she increase her planting of silver bells? How much should she
increase her planting of cockle shells?
(c) If Mary had only 144 square feet in her garden, how many cockle shells would
she grow?
(d) If Mary grows both silver bells and cockle shells, what must be the range for
Mary’s income?
3. Casper consumes cocoa and cheese. He has an income of $16. Cocoa is sold in an
unusual way. There is only one supplier and the more cocoa one buys from him,
the higher the price one has to pay per unit. In fact,
units of cocoa will cost
Casper a total of
dollars. Cheese is sold in the usual way at a price of $2 per
unit. Casper’s budget equation, therefore, is
2
16 where
is his
consumption of cocoa and
is his consumption of cheese. Casper’s utility
function is
,
3
.
(a) On the graph, draw Casper’s budget line and sketch two or three of his
indifference curves.
(b) How many units of cocoa and cheese does Casper demand, respectively?
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4. Agatha must travel on the Orient Express from Istanbul to Paris. The distance is
1,500 miles. A traveler can choose to make any fraction of the journey in a
first-class carriage and travel the rest of the way in a second-class carriage. The
price is 10 cents a mile for a second-class carriage and 20 cents a mile for a
first-class carriage. Agatha much prefers first-class to second-class travel, but
because of a misadventure in an Istanbul bazaar, she has only $200 left with which
to buy her tickets. Luckily, she still has her toothbrush and a suitcase full of
cucumber sandwiches to eat on the way. Agatha plans to spend her entire $200 on
her tickets for her trip. She will travel first class as much as she can afford to, but
she must get all the way to Paris, and $200 is not enough money to get her all the
way to Paris in first class.
(a) On the graph below, use red ink to show the locus of combinations of firstand second-class tickets that Agatha can just afford to purchase with her $200.
Use blue ink to show the locus of combinations of first- and second-class
tickets that are sufficient to carry her the entire distance from Istanbul to Paris.
Locate the combination of first- and second-class miles that Agatha will
choose on your graph and label it A.
(b) Let m1 be the number of miles she travels by first-class coach and m2 be the
number of miles she travels by second-class coach. Write down two equations
that you can solve to find the number of miles she chooses to travel by
first-class coach and the number of miles she chooses to travel by second-class
coach.
(c) How many miles does she choose to travel by second-class coach?
(d) Just before she was ready to buy her tickets, the price of second-class tickets
fell to $.05 while the price of first-class tickets remained at $.20. On the graph
that you drew above, use pencil to show the combinations of first-class and
second-class tickets that she can afford with her $200 at these prices. On your
graph, locate the combination of first-class and second-class tickets that she
would now choose. (Remember, she is going to travel as much first-class as
she can afford to and still make the 1,500 mile trip on $200.) Label this point
B. How many miles does she travel by second class now? (Hint: For an exact
solution you will have to solve two linear equations in two unknowns.) Is
second-class travel a normal good for Agatha? Is it a Giffen good for her?
5. We continue with the adventures of Agatha, from the previous problem. Just after
the price change from $.10 per mile to $.05 per mile for second-class travel, and
just before she had bought any tickets, Agatha misplaced her handbag. Although
she kept most of her money in her sock, the money she lost was just enough so
that at the new prices, she could exactly afford the combination of first- and
second-class tickets that she would have purchased at the old prices. How much
money did she lose? On the graph you started in the previous problem, use black
ink to draw the locus of combinations of first- and second-class tickets that she
can just afford after discovering her loss. Label the point that she chooses with a C.
How many miles will she travel by second class now?
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(a) Finally, poor Agatha finds her handbag again. How many miles will she travel
by second class now (assuming she didn’t buy any tickets before she found her
lost handbag)? When the price of secondclass tickets fell from $.10 to $.05,
how much of a change in Agatha’s demand for second-class tickets was due to
a substitution effect? How much of a change was due to an income effect?
6. Mario has a small garden where he raises eggplant and tomatoes. He consumes
some of these vegetables, and he sells some in the market. Eggplants and tomatoes
are perfect complements for Mario, since the only recipes he knows use them
together in a 1:1 ratio. One week his garden yielded 30 pounds of eggplant and 10
pounds of tomatoes. At that time the price of each vegetable was $5 per pound.
(a) What is the monetary value of Mario’s endowment of vegetables?
(b) On the graph, use blue ink to draw Mario’s budget line. How many pounds of
tomatoes and eggplant, respectively, does Mario end up consuming? Draw the
indifference curve through the consumption bundle that Mario chooses and
lable this bundle A.
(c) Suppose that before Mario makes any trades, the price of tomatoes rises to $15
a pound, while the price of eggplant stays at $5 a pound. What is the value of
Mario’s endowment now? Draw his new budget line, using red ink. How many
pounds of tomatoes and eggplants will he will now choose to consume?
(d) Suppose that Mario had sold his entire crop at the market for a total of $200,
intending to buy back some tomatoes and eggplant for his own consumption.
Before he had a chance to buy anything back, the price of tomatoes rose to $15,
while the price of eggplant stayed at $5. Draw his budget line, using pencil or
black ink. How many pounds of tomatoes and eggplants will he will now
choose to consume?
(e) Assuming that the price of tomatoes rose to $15 from $5 before Mario made
any transactions, what was the change in the demand for tomatoes due to the
substitution effect? What was the change in the demand for tomatoes due to
the ordinary income effect? What was the change in the demand for tomatoes
due to the endowment income effect? What was the total change in the
demand for tomatoes?
7. Laertes has an endowment of $20 each period. He can borrow money at an
interest rate of 200%, and he can lend money at a rate of 0%. (Note: If the interest
rate is 0%, for every dollar that you save, you get back $1 in the next period. If the
interest rate is 200%, then for every dollar you borrow, you have to pay back $3 in
the next period.)
(a) Use blue ink to illustrate his budget set on a graph.
(b) Laertes could invest in a project that would leave him with m1= 30 and m2 =
15. Besides investing in the project, he can still borrow at 200% interest or
lend at 0% interest. Use red ink to draw the new budget set in the graph above.
Would Laertes be better off or worse off by investing in this project given his
possibilities for borrowing or lending? Or can’t one tell without knowing
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something about his preferences? Explain.
(c) Consider an alternative project that would leave Laertes with the endowment
m1 = 15, m2 = 30. Again suppose he can borrow and lend as above. But if he
chooses this project, he can’t do the first project. Use pencil or black ink to
draw the budget set available to Laertes if he chooses this project. Is Laertes
better off or worse off by choosing this project than if he didn’t choose either
project? Or can’t one tell without knowing more about his preferences?
Explain.
8. Lolita, an intelligent and charming Holstein cow, consumes only two goods, cow
feed (made of ground corn and oats) and hay. Her preferences are represented by
the utility function U(x, y) = x−x2/2+y, where x is her consumption of cow feed
and y is her consumption of hay. Lolita has been instructed in the mysteries of
budgets and optimization and always maximizes her utility subject to her budget
constraint. Lolita has an income of $m that she is allowed to spend as she wishes
on cow feed and hay. The price of hay is always $1, and the price of cow feed will
be denoted by p, where 0  p  1 .
(a) Write Lolita’s inverse demand function for cow feed.
(b) If the price of cow feed is p and her income is m, how much hay does Lolita
choose? (Hint: The money that she doesn’t spend on feed is used to buy hay.)
(c) Plug these numbers into her utility function to find out the utility level that she
enjoys at this price and this income.
(d) Suppose that Lolita’s daily income is $3 and that the price of feed is $.50.
What bundle does she buy? What bundle would she buy if the price of cow
feed rose to $1?
(e) How much money would Lolita be willing to pay to avoid having the price of
cow feed rise to $1? What variation is this amount known as?
(f) Suppose that the price of cow feed rose to $1. How much extra money would
you have to pay Lolita to make her as well-off as she was at the old prices?
What variation is this amount known as? Which is bigger, the compensating or
the equivalent variation, or are they the same?
9. The demand function for yo-yos is D(p,M) = 4 − 2p + 0.01M, where p is the price
of yo-yos and M is income. If M is 100 and p is 1,
(a) What is the income elasticity of demand for yo-yos?
(b) What is the price elasticity of demand for yo-yos?
10. The demand function for football tickets for a typical game at a large midwestern
university is D(p) = 200,000 −10,000p. The university has a clever and avaricious
athletic director who sets his ticket prices so as to maximize revenue. The
university’s football stadium holds 100,000 spectators.
(a) Write down the inverse demand function.
(b) Write expressions for total revenue and marginal revenue as a function of the
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number of tickets sold.
(c) What price will generate the maximum revenue? What quantity will be sold at
this price?
(d) At this quantity, what is marginal revenue? At this quantity, what is the price
elasticity of demand? Will the stadium be full?
(e) A series of winning seasons caused the demand curve for football tickets to
shift upward. The new demand function is q(p) = 300,000 −10,000p. What is
the new inverse demand function?
(f) Write an expression for marginal revenue as a function of output.
(g) Ignoring stadium capacity, what price would generate maximum revenue?
What quantity would be sold at this price?
(h) As you noticed above, the quantity that would maximize total revenue given
the new higher demand curve is greater than the capacity of the stadium.
Clever though the athletic director is, he cannot sell seats he hasn’t got. He
notices that his marginal revenue is positive for any number of seats that he
sells up to the capacity of the stadium. Therefore, in order to maximize his
revenue, how many tickets should he sell? And at what price?
(i) When he does this, what is his marginal revenue from selling an extra seat?
What is the elasticity of demand for tickets at this price quantity combination?
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