PAM200 – Section 5
Problem 5.3
a)
,
%
/
%
/
Q I
Q
Q Q
Q
I
I
I I
I
Q
ε
∆
∆
∆
=
=
=
∆
∆
∆
I
and
Q
must be greater than zero.
In addition, assume income increases,
i.e.,
0
I
∆
.
If the good is inferior, then
0
Q
∆
<
.
Thus, the first term (
/
)
0
Q
I
∆
∆
<
and the second term ( /
)
0
I Q
.
Multiplying these two terms together implies
,
0
Q I
<
.
Inferior goods have a negative income elasticity of demand.
b)If income elasticity of demand is negative then
,
0
Q I
Q
I
I
Q
∆
=
<
∆
.
Since
I
and
Q
must be greater than zero, for
,
Q I
to be negative, we must have
0
Q
I
∆
<
∆
.
This can only happen if either a)
0
Q
∆
<
and
0
I
∆
or b)
0
Q
∆
and
0
I
∆ <
.
In both instances, the change in quantity
demanded moves in the opposite direction as the change in income implying the good is inferior.
Problem 5.6
a)
Karl’s optimal bundle will always be such that 2
H
= 3
B
. If this were not true then he could decrease the consumption of one
of the two goods, staying at the same level of utility and reducing expenditure. Also, at the optimal bundle, it must be true that
I
B
P
H
P
B
H
=
+
. Substituting the first condition into the second we get
I
P
P
B
B
H
=
+
)
5
.
1
(
which implies that the
demand curve for beer is given by,
)
5
.
1
(
B
H
P
P
I
B
+
=
b)
You can answer this just by looking at the demand curve. Because it has a larger coefficient, the price of hamburgers affects
the demand for beer more than the price of beer. A one dollar increase in
H
P
decreases demand for beer more than a one dollar
increase in
B
P
.
Problem 5.16
a)Using the tangency condition,
4
=
x
y
, and the budget constraint,
120
4
=
+
y
x
, Lou’s initial optimum is the basket (
x
,
y
) =
(15, 60) with a utility of 900.
b)First we need the decomposition basket. This would satisfy the new tangency condition,
3
=
x
y
and would give him as much
utility as before, i.e.
900
=
xy
. This gives
)
3
30
,
3
10
(
)
,
(
=
y
x
or approximately (17.3,51.9). Now we need the final
basket, which satisfies the same tangency condition as the decomposition basket and also the new budget constraint:
.
120
3
=
+
y
x
Together, these conditions imply that (
x
,
y
) = (20, 60). The substitution effect is therefore 17.3 – 15 = 2.3, and
the income effect is 20 – 17.3 = 2.7.
c)The compensating variation is the amount of income Lou would be willing to give up after the price change to maintain the level
of utility he had before the price change. This equals the difference between the consumer’s actual income, $120, and the income
needed to buy the decomposition basket at the new prices. This latter income equals: 3*17.3 + 1*51.9 = 103.8. The compensating
variation thus equals 120 – 103.8 = $16.2.
d)The equivalent variation is the amount of income that Lou would need to be given