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# Varian Workouts (Solutions)

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```Chapter 1 NAME
The Market
Introduction. The problems in this chapter examine some variations on
the apartment market described in the text. In most of the problems we
work with the true demand curve constructed from the reservation prices
of the consumers rather than the \smoothed&quot; demand curve that we used
in the text.
Remember that the reservation price of a consumer is that price
where he is just indi\ufffderent between renting or not renting the apartment.
At any price below the reservation price the consumer will demand one
apartment, at any price above the reservation price the consumer will de-
mand zero apartments, and exactly at the reservation price the consumer
will be indi\ufffderent between having zero or one apartment.
You should also observe that when demand curves have the \stair-
case&quot; shape used here, there will typically be a range of prices where
supply equals demand. Thus we will ask for the the highest and lowest
price in the range.
1.1 (3) Suppose that we have 8 people who want to rent an apartment.
Their reservation prices are given below. (To keep the numbers small,
think of these numbers as being daily rent payments.)
Person = A B C D E F G H
Price = 40 25 30 35 10 18 15 5
(a) Plot the market demand curve in the following graph. (Hint: When
the market price is equal to some consumer i\u2019s reservation price, there
will be two di\ufffderent quantities of apartments demanded, since consumer
i will be indi\ufffderent between having or not having an apartment.)
2 THE MARKET (Ch. 1)
0 1 2 3 4 5 6 7 8
10
20
30
40
60
50
Price
Apartments
(b) Suppose the supply of apartments is \ufffdxed at 5 units. In this case
there is a whole range of prices that will be equilibrium prices. What is
the highest price that would make the demand for apartments equal to 5
units? \$18.
(c) What is the lowest price that would make the market demand equal
to 5 units? \$15.
(d) With a supply of 4 apartments, which of the people A{H end up
getting apartments? A, B, C, D.
(e) What if the supply of apartments increases to 6 units. What is the
range of equilibrium prices? \$10 to \$15.
1.2 (3) Suppose that there are originally 5 units in the market and that
1 of them is turned into a condominium.
(a) Suppose that person A decides to buy the condominium. What will
be the highest price at which the demand for apartments will equal the
supply of apartments? What will be the lowest price? Enter your an-
swers in column A, in the table. Then calculate the equilibrium prices of
apartments if B, C, : : : , decide to buy the condominium.
NAME 3
Person A B C D E F G H
High price 18 18 18 18 25 25 25 25
Low price 15 15 15 15 18 15 18 18
(b) Suppose that there were two people at each reservation price and 10
apartments. What is the highest price at which demand equals supply?
18. Suppose that one of the apartments was turned into a condo-
minium. Is that price still an equilibrium price? Yes.
1.3 (2) Suppose now that a monopolist owns all the apartments and that
he is trying to determine which price and quantity maximize his revenues.
(a) Fill in the box with the maximum price and revenue that the monop-
olist can make if he rents 1, 2, : : :, 8 apartments. (Assume that he must
charge one price for all apartments.)
Number 1 2 3 4 5 6 7 8
Price 40 35 30 25 18 15 10 5
Revenue 40 70 90 100 90 90 70 40
(b) Which of the people A{F would get apartments? A, B, C, D.
(c) If the monopolist were required by law to rent exactly 5 apartments,
what price would he charge to maximize his revenue? \$18.
(d) Who would get apartments? A, B, C, D, F.
(e) If this landlord could charge each individual a di\ufffderent price, and he
knew the reservation prices of all the individuals, what is the maximum
revenue he could make if he rented all 5 apartments? \$148.
(f) If 5 apartments were rented, which individuals would get the apart-
ments? A, B, C, D, F.
1.4 (2) Suppose that there are 5 apartments to be rented and that the
city rent-control board sets a maximum rent of \$9. Further suppose that
people A, B, C, D, and E manage to get an apartment, while F, G, and
H are frozen out.
4 THE MARKET (Ch. 1)
(a) If subletting is legal|or, at least, practiced|who will sublet to whom
in equilibrium? (Assume that people who sublet can evade the city rent-
control restrictions.) E, who is willing to pay only
\$10 for an apartment would sublet to F,
who is willing to pay \$18.
(b) What will be the maximum amount that can be charged for the sublet
payment? \$18.
(c) If you have rent control with unlimited subletting allowed, which of
the consumers described above will end up in the 5 apartments? A,
B, C, D, F.
(d) How does this compare to the market outcome? It\u2019s the
same.
1.5 (2) In the text we argued that a tax on landlords would not get
passed along to the renters. What would happen if instead the tax was
imposed on renters?
(a) To answer this question, consider the group of people in Problem 1.1.
What is the maximum that they would be willing to pay to the landlord
if they each had to pay a \$5 tax on apartments to the city? Fill in the
box below with these reservation prices.
Person A B C D E F G H
Reservation Price 35 20 25 30 5 13 10 0
(b) Using this information determine the maximum equilibrium price if
there are 5 apartments to be rented. \$13.
(c) Of course, the total price a renter pays consists of his or her rent plus
the tax. This amount is \$18.
(d) How does this compare to what happens if the tax is levied on the
landlords? It\u2019s the same.
Chapter 2 NAME
Budget Constraint
Introduction. These workouts are designed to build your skills in de-
scribing economic situations with graphs and algebra. Budget sets are a
good place to start, because both the algebra and the graphing are very
easy. Where there are just two goods, a consumer who consumes x1 units
of good 1 and x2 units of good 2 is said to consume the consumption bun-
dle, (x1; x2). Any consumption bundle can be represented by a point on
a two-dimensional graph with quantities of good 1 on the horizontal axis
and quantities of good 2 on the vertical axis. If the prices are p1 for good 1
and p2 for good 2, and if the consumer has income m, then she can a\ufffdord
any consumption bundle, (x1; x2), such that p1x1+p2x2 \ufffd m. On a graph,
the budget line is just the line segment with equation p1x1 + p2x2 = m
and with x1 and x2 both nonnegative. The budget line is the boundary
of the budget set. All of the points that the consumer can a\ufffdord lie on
one side of the line and all of the points that the consumer cannot a\ufffdord
lie on the other.
If you know prices and income, you can construct a consumer\u2019s bud-
get line by \ufffdnding two commodity bundles that she can \just a\ufffdord&quot; and
drawing the straight line that runs through both points.
Example: Myrtle has 50 dollars to spend. She consumes only apples and
bananas. Apples cost 2 dollars each and bananas cost 1 dollar each. You
are to graph her budget line, where apples are measured on the horizontal
axis and bananas on the vertical axis. Notice that if she spends all of her
income on apples, she can a\ufffdord 25 apples and no bananas. Therefore
her budget line goes through the point (25; 0) on the horizontal axis. If
she spends all of her income on bananas, she can a\ufffdord 50 bananas and
no apples. Therfore her budget line also passes throught the point (0; 50)
on the vertical axis. Mark these two points on your graph. Then draw a
straight line between them. This is Myrtle\u2019s budget line.
What if you are not told prices or income, but you know two com-
modity bundles that the consumer can just a\ufffdord? Then, if there are just
two commodities, you know that a unique line can be drawn through two
points, so you have enough information to draw the budget line.
Example: Laurel consumes only ale and bread.```