Econ 11  Additional Problems
The problems you
fi
nd below were used in last year’s Econ 11 course as homework prob
lems and on the midterm.
You may
fi
nd them useful in preparing for this year’s exam.
Solutions to these problems will be posted separately.
1. John is a student at university who spends most of the stipend he receives from his
parents on meals
m
, books
b
and on movie tickets
t
. His preferences are given by the utility
function
U
(
m, b, t
) = 2 log
m
+ log
b
+ log
t
(
i
)
If the price of a meal is 5$, the price of a book is 25$ and the price of a ticket is 10$,
and John’s stipend is 400$ per month, how will he allocate his stipend each month?
(
ii
)
John’s parents are concerned that over all the movies that he watches, he does not
read enough books. His parents consider reducing his stipend by 100$. By how much will
such a change in the stipend reduce the number of movies John watches?
How will his
consumption of meals and books be a
ff
ected? Do you think this is an e
ff
ective measure to
reduce the time John spends at the movies?
(
iii
)
Ultimately, John’s parents are most concerned that John eats well and reads enough
books for his exams, and do not care about his movie consumption. Their preferences for
John can be expressed by the utility function
V
(
m, b
) = log
m
+ log
b
If they could dictate how John should allocate his stipend, how would they like him to
spend his money?
(
iv
)
The parents realize they cannot monitor how their son spends his stipend, but they
nevertheless want to in
fl
uence his behavior. They decide to reduce his stipend to some new
amount
X
, and they decide to subsidize each purchase of a book by an additional
Y
dollars.
Rewrite John’s budget line for his parent’s new policy, and solve the new optimization
problem for a given allowance
X
and book subsidy
Y
.
(
v
)
Derive John’s optimal purchases of
m
,
b
, and
t
as a function of
X
and
Y
. What are
his purchases, if
X
= 200
and
Y
= 20
. What is the total cost to his parents? Show that
this new agreement reduces John’s utility, but increases his parent’s utility, relative to the
previous stipend of
400
.
(
vi
)
Suppose that John’s parents want to spend a total of 400$ per month for the stipend
and the book subsidy.
Use John’s optimal choice of
b
(
X, Y
)
to
fi
nd the parental budget
constraint between the allowance
X
and the subsidy
Y
; that is the set of values
X
,
Y
, for
which, given John’s choice
b
(
X, Y
)
, the parents spend a total amount of exactly
400
. Sketch
the parental budget constraint. Explain, why this budget constraint is not a straight line.
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 Spring '08
 cunningham
 Utility maximization problem, utility level, San Serrife

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