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Attached are some practice problems from my econ class, if you could provide me with the steps and answers that would be great. Thanks.

EC 3306: Practice Questions 1. (20 points) The utility function a 1 u ( x1 , x 2 ) x x 1 a 2 , where Douglas’ preferences for tickets to baseball…

Attached are some practice problems from my econ class, if you could provide me with the steps and answers that would be great. Thanks. 

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EC 3306: Practice Questions
1. (20 points) The utility function
a
a
x
x
x
x
u


1
2
1
2
1
)
,
(
, where
1
0


a
, represents Cobbie
Douglas’ preferences for tickets to baseball games (good
1
x
) and to football games (good
2
x
).
Suppose that Cobbie’s income is
m
, and that the ticket prices are respectively
1
p
and
2
p
.
(a) (5 points) Write down Cobbie’s budget constraint and MRS of his utility function.
(b) (5 points) Derive the individual demand functions for baseball and football tickets using
answers to (a). Say whether Cobbie treats goods as normal or inferior, ordinary or Giffen, and
whether they are complements or substitutes of one another.
(c) (5 points) How would you describe in words the preferences corresponding to
0

a
and
1

a
? Does your explanation agree with the demands that you found for these cases?
Explain.
(d) (5 points) Assume
2
1

a
and
m
=100 and
2
p
=10. Draw the demand curve for good
1
x
. Next
let
m
=100 and
1
p
=20 and draw the cross demand curve for good
1
x
. What happens to these two
curves if
m
increases to 200?
2. (20 points) Mr. Leontief always drinks each cup of coffee (good
1
x
) with one spoon of sugar
(good
2
x
). Therefore, his preferences for the two goods can be represented by the utility function


2
1
2
1
,
2
min
)
,
(
x
x
x
x
u

. Suppose that Mr. Leontief’s income (m) is 100 dollars, the price per
cup of coffee is 10 dollars and the price per spoon of sugar is 7.5 dollars.
(a) (5 points) What is his optimal consumption plan?
(b) (5 points) Suppose that, due to new import taxes, the prices go up to 15 dollars and 17.5
dollar, respectively. Find his new optimal choice. How much will he be paying in taxes?
(c) (5 points) Suppose prices go back to the initial levels (10 dollars and 7.5 dollar, respectively),
but that Mr. Leontief’s income is 150 dollars
(d) (5 points) Derive his demand functions for cups of coffee and spoons of sugar. Use
m
for
income, p
1
and p
2
for prices of coffee and sugar, respectively. Are the goods normal or inferior,
ordinary or Giffen, substitutes or complements?
3. (15 points) Sammy and Jimmy are twin brothers. Their weekly allowance is 2 dollars for each.
Sammy’s preferences for baseball card (good
1
x
) and “famous economists” cards (good
2
x
) are
represented by the utility function
2
1
2
1
2
)
,
(
x
x
x
x
u


. Suppose that the unit price of both
goods is 1 dollar.

View the Answer

(a) (5 points) Solve for Sammy’s optimal choice.
(b) (5 points) Suppose
1
p
rises to 2 dollars. What is his new optimal consumption plan?
(c) (5 points) Jimmy’s preferences are represented by
2
2
1
2
1
)
2
(
)
,
(
x
x
x
x
u


. Answer parts (a)
and (b) for Jimmy. Are answers different? Why or why not?
4. (15 points) “Ernie and Bert, Inc.” sells cookies (good
1
x
) and bananas (good
2
x
). They are
offering the following deal to their customers. Although the price of bananas is fixed at 1 dollar
per unit, the first three cookies that a consumer buys are free. After the third cookie, the price per
cookie is also 1 dollar. Cookie Monster’s preferences for both goods are represented by
)
3
(
)
,
(
2
1
2
1


x
x
x
x
u
and his income is 5 dollars.
(a) (10 points) Draw Cookie Monster’s budget line.
(b) (5 points) Find his optimal consumption bundle.
5. (30 points) Say whether the following is true or false. Explain it briefly (For a true statement,
an explanation is not necessary. For a false statement, no explanation, no credits!)
(a) (10 points) For Quasi whose utility is defined by:
2
1
2
1
ln
)
,
(
x
x
x
x
u


, he consumes a o
amount of
x
1
and
positive amount of x
2
when his income level is very low. He begins to
consume a positive amount of
x
1
, when he has enough income. More specifically, his demand
for
x
1
is positive if and only if his income (m) is greater than p
1
.
(b) (10 points) If a consumer has homothetic preferences and income is scaled up or down by any
amount t (t>0), demand bundle scales up or down by the same amount t.
(c) (10 points) When prices are ($2, $2), Ms. Consumer chooses the bundle (10,1), and when
prices are ($1, $2), she chooses the bundle (5, 4). Then, she satisfies the weak axiom of
revealed reference.

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