Test 1 solution sketches
Average: 42.11 points (75.2%)
5 tests with score 56
#1: Dianne

Dianne’s utility (U) is
the fourth root of her
4
income (I), i.e. U = 𝐼.
She has a 16% chance
of receiving an income
of $6,250,000 and an
84% chance of
receiving an income of
$810,000.

(a) (3 points) What is the
expected income of this
gamble?


(b) (3 points) What is the
expected utility of this
gamble?


0.16($6,250,000) +
0.84($810,000) =
$1,680,400
0.16(6,250,000)0.25 +
0.84(810,000)0.25 = 33.2
(c) (4 points) What is the
certainty equivalent of
this gamble?

CE = EU4 = 33.24 =
$1,214,933
#2: The Hollymetal Bowl

The Hollymetal Bowl invites
top acts for concerts, but
people that live for miles
around also enjoy the
music. Assume Q is the
number of hours of concerts
each year. Total cost for
these concerts comes out to
TC = Q2 +5Q. Demand,
which reflects marginal
private benefit of
consumption, is represented
by the equation P = 500 –
2Q. Since people that live in
neighboring houses also
enjoy the concerts, there is
an external benefit of EB =
50Q.

(a) (5 points) What is the
actual output by the
Hollymetal Bowl if external
benefits are not taken into
account. Assume a
competitive market.





Set MB = MC (excluding
external benefit)
MB comes from the demand
curve
MC = dTC/dQ = 2Q + 5
500 – 2Q = 2Q + 5
Q = 123.75
#2: The Hollymetal Bowl

(b) (5 points) What is the socially efficient
output of concerts?

Set SMB = MC





Note that MEB = dEB/dQ = 50
SMB = (500 – 2Q) + 50
SMB = 550 – 2Q
550 – 2Q = 2Q + 5
Q = 136.25
#3: Lester and Jacqueline

Lester and Jacqueline
are wondering what to
do with educating their
child, Hillary. The
family’s utility function
is U(x,y) = xy3, where x
represents $1000s
spent on educating
Hillary, and y
represents $1000s
spent on everything
else. The household is
able to spend $50,000.

(a) (5 points) How much
money should be spent
on education if there is
no public education
available?



max xy3 such that x+y=50
Substitute to get
max (50 – y)y3, or
max 50y3 – y4
FOC: 150y2 – 4y3




Set equal to zero to get
y = 0 or y = 37.5
Note that y = 0 gives a
minimum for utility
So choose y = 37.5
Choose x = 12.5
#3: Lester and Jacqueline




(b) (6 points) Suppose that Lester and Jacqueline have an additional
option here: Publicly-provided education. Lester and Jacqueline have
two choices:
Choose public education, in which $5,000 of taxpayer money is spent to
educate Hillary. Note that Lester and Jacqueline pay nothing for
education here, but cannot spend any of their money to educate Hillary.
Also note that $50,000 of household money is available to be spent
whether or not public schooling is chosen.
Spend as much household money as they want to educate Hillary. If
they choose this option, they are not allowed to use any taxpayer
money for educating Hillary.
What should Lester and Jacqueline do to maximize family utility? Justify
your answer.



No public education: Choose answer from (a) to get a utility of 659,180
With public education: Choose x= 5 and y = 50 to get a utility of 625,000
Since the higher utility is with no public education, choose this, and spend a value
of x = 12.5, y = 37.5
#4: Pismo Valley Heights

Two hundred people live in
the land of Pismo Valley
Heights. Eighty of these
people have individual
demand curves for a good
given by P = 200 – 4Q,
where P is price and Q is
number of units. One
hundred twenty of these
people have individual
demand curves for the
same good given by
P = 100 – Q. The marginal
cost of this good is 30.

(a) (4 points) How many units
of the good are purchased in
total (i.e. the total quantity
purchased by all 200 people)
if this is a private good?
Assume a competitive
market.



80 people consume Q such
that 200 – 4Q = 30  Q = 42.5
120 people consume Q such
that 100 – Q = 30  Q = 70
Total consumption

80*42.5 + 120*70 = 11,800
#4: Pismo Valley Heights

(b) (6 points) How many units of the good are
provided in an efficient outcome if this is a
public good?

Vertical summation

For Q between 0-50




WTPtotal = 80(200 – 4Q) + 120(100 – Q) = 28,000 – 440Q
Set WTPtotal = MC
 28,000 – 440Q = 30  Q = 63.568
DO NOT CHOOSE THIS (Q is not between 0-50)
For Q between 50-100 (note that 80 people have WTP = 0
here)


WTPtotal = 120(100 – Q) = 12,000 – 120Q
Set WTPtotal = MC
 12,000 – 120Q = 30  Q = 99.75 (This is the answer!)
#5: 2 routes between Goldville & Silverville

5. There are two routes to
travel between Goldville and
Silverville. One route is
Wide Back Road, in which
anybody traveling on this
route has a travel time of 45
minutes. The other route is
Narrow Mountain Highway,
in which the travel time
depends on the number of
travelers on the highway.
The travel time on the
highway is 21 + T, with T
representing the total
number of drivers on the
highway.

(a) (3 points) If there is no
regulation on travel between
Goldville and Silverville, what
is equilibrium if there are 100
travelers total? Make sure to
describe what happens on
both routes.



Set 21 + T = 45
T = 24
# of drivers on Wide Back Rd.
is 100 – 24, or 76
#5: 2 routes between Goldville & Silverville

(b) (4 points) Suppose that someone on the
Goldville town council proposes to improve
Narrow Mountain Highway, such that the travel
time on the highway is 21 + 0.4T. How does this
change equilibrium travel times relative to part
(a)? Use math and/or an explanation of 40
words or less to justify your answer.


Set 21 + 0.4T = 45  T = 60  # of drivers on Wide
Back Rd. = 100 – 60 = 40
Note that equilibrium travel times are the same on
each route after the expansion (21 + 0.4*60 = 45)
#5: 2 routes between Goldville & Silverville

(c) (4 points) Once again use the information
before part (a) to solve this part: What is efficient
if you are trying to minimize the total number of
minutes of travel time of all travelers? Make sure
to describe what happens on both routes.



min 45(100 – T) + (21 + T)T
min 4500 – 24T + T2
Take the FOC and set it equal to 0


-24 + 2T = 0  T = 12
# of drivers on Wide Back Rd. = 100 – 12 = 88
#5: 2 routes between Goldville & Silverville

(d) (4 points) Once again use the information before part
(a) to solve this part: Suppose that you know that everyone
traveling between Goldville and Silverville have values of
time of $120 per hour. How much of a toll would be
charged on one or both routes in order to minimize the total
number of minutes of travel time of all travelers in
equilibrium?


Value of time is $120 per hour, or $2 per minute
Total cost to travel on either route should be the same


This includes both tolls and travel time cost
Set toll + 2*33 = 2*45  toll = $24


Note that toll is only on the highway
Another way to think of this is that you need to charge a toll
equivalent to 12 minutes of travel time on the highway

Since value of time is $2/min., 12 min. of time = $24 in tolls
Level of difficulty for each question

Easy



1a, 1b, 1c, 2a, 2b, 4a,
5a, 5b
31 points
Medium


3a, 3b
11 points

Hard



4b, 5c
10 points
Hard to very hard


5d
4 points