THE GEORGE WASHINGTON UNIVERSITY
Department of Economics
A. Yezer
Answers to 4
th
Problem Set in Econ 11-10
Fall 2007
1a. Assume that Jahangir currently consumes 10 apples per month and that his monthly marginal
utility for apples is given by MU
A
= 20 - A, where A is the number of apples consumed per month and
MU is marginal utility.
Assume further that his marginal utility of grapes and bananas are given by MU
G
= 40 - 4G, and MU
B
= 60 - 6B, where G and B are in pounds per month.
You are told that apples
cost $1 each, grapes cost $2 per pound and bananas cost $3 per pound.
If Jahangir currently
consumes 10 apples per month, how many pounds of grapes and bananas does he consume per month?
How would your answer change if Jahangir consumed 5 apples per month?
In this problem you are to apply the
golden rule
for utility-maximization from
consumer's demand theory which states that, for all goods consumed, the marginal utility of
the last dollar spent on each good must be equal.
Now the marginal utility of the last dollar
spent on a good is the quotient of marginal utility divided by price.
In this case, we know that
MU
A
= 20 - A, and P
A
= 1, so MU
A
/P
A
= (20 - A)/1 = 20 - A, and we see that it is a simple
decreasing function of A.
If Abhijit consumes 10 apples per month, then MU
A
/P
A
= 20 - 10 =
10.
We also know that MU
B
= 60 - 6B, and P
B
= 3, so MU
B
/P
B
= (60 - 6B)/3 = 20 - 2B.
Furthermore, MU
G
= 40 - 4G and P
G
= 2, so MU
G
/P
G
= (40 - 4G)/2 = 20 - 2G.
Given that Abhijit consumes 10 apples, we know that the marginal utility of the last
dollar spent on apples is 10 as established above.
Therefore, the marginal utility of the last
dollar spent on bananas and grapes must also be 10 for him to maximize utility.
How many B
and G does it take to achieve this result?
Setting 10 = 20 - 2G and find that G = (10 - 20)/-2 =
-10/-2 = 5 pounds of grapes and setting 10 = 20 - 2B, find that B = 5 pounds of banannas is the
combination of B and G that maximizes utility!
1b. Assume that biologists have created a new fruit, the lemopear, by crossing lemons with
pears and that Jahangir 's MU for lemopears is given by MU
L
= 10 - L.
This new creation has been
priced at $5 each.
How many lemopears will
Jahangir consumer per month?
Let's solve for the relation between the number of lemopears and marginal utility of
the last dollar spent on L as follows: MU
L
/P
L
= (10 - L)/5 = 2 - 0.2L.
But
Jahangir
already
gets MU/P = 10 from apples.
But even when L = 0 MU
L
/P
L
= 2, so there is no way that
lemopears can generate sufficient marginal utility per dollar of expenditure for them to be
attractive to Jahangi
r
.
He maximizes utility by consuming 0 lemopears.
This is called a
"corner solution" and we all know lots of goods that we don't consume.
For these goods, the
marginal utility of the first dollar of expenditure is low enough so that it does not attract us
compared to alternatives.
Likely most of you would not be attracted by a fruit produced by
crossing a lemon and a pear - a silly, proposition that will never be undertaken unless