Department of Economics
W3211
Columbia University
Fall 2011
SO LUTIONS T O
Probl
e
m S
e
t 3
Int
e
rm
e
diat
e
Mi
c
ro
ec
onomi
c
s
Prof
.
S
e
yhan E Arkona
c
1
.
S
teve’s utility for socks
(q
1
) and other goods (q
2
) is given byU(q
1
,q
2
) = 10q
1
.1
q
2
.9
The
price of the composite good is p
2
=1 and the price of a pair of socks is p
1
=2. Steve’s
income is Y=100.
Every year, Steve’s mom buys him 20 pairs of socks. Find the
equivalent variation of the gift.
What is the difference between the cost of the gift and
the equivalent valuation cash amount?
Answer: First we must find the bundle that Steve receives after the gift. The Lagrangian
with an income of 140 suggests that q
1
=7. However, we know that Steve must consume
at least 20 socks (from the gift), so his bundle is the corner solution (20,100).The utility
of the bundle (20,100) is 851.
When MRS is set equal to MRT we find that q
2
= 18q
1
. We then plug this value for q
2
into the utility function and solve for the bundle that gives the same utility as the gift
bundle.
851= 10q
1
.1
(18q
1
)
.9
This bundle is (6.3,113.4).
The cost of this bundle is $126, so the EV=26. The difference between the cost of the gift
and the EV is 4026=14.
The gift cost $40 whereas Steve would be just as happy with $26 in cash.
2
.
For each of the following statements, define all of the underlined terms
.
Then, explain
why the statement is true or false
.
a
.
If a consumer views two goods as perfect substitutes
then their optimal choice will be
a corner solution
.
b
.
The substitution effect
from a price increase states that the consumer will always
choose a smaller amount of that good to consume.
However, the income effect
states
that consumption can move in either direction.
c
.
Suppose Alf and Bo haveconvex indifference curves
. Alf likes units of "X’ more
than units of "Y’ but Bo likes units of "Y’ much more than units of "X’. Then, in the
optimum, Alf’s
marginal rate of substitution
will be different from Bo’s even if they
face the same prices.
d
.
All Giffen goods
are normal goods
, but not all normal goods are Giffen goods.
e
.
Economists assume that preferences are ordinal
.
This implies that given two utility
functions and one is a monotonic transformation
of the other, then they represent the
same preferences over bundles of goods.
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Answer:
a
.
Perfect substitutes are goods for which at any point you will trade one good in the
same proportion for the other.
The specific utility function is U(x,y)=ax+by, and the
ICs are linear throughout.
A corner solution is when the optimal bundle is all of one
good and none of the other (i.e. the solution is at the intercept of the BC and one of
the axis). True, since the MRS is constant for p.s.’s, the o
ptimal bundle will occur at a
corner. False, if they overlap.
b
.
The substitution effect is the change in consumption that occurs following a price
change based purely on the fact that relative prices have changed while
holding
utility fix
e
d
. The income effect is the change in consumption following a price
change that results from the person feeling richer or poorer, while
holding r
e
lativ
e
pri
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 Fall '09
 Elmes
 Microeconomics, Utility

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