University of California, Davis
Date:
August 28, 2008
Department of Economics
Time:
5 hours
Microeconomics
Reading Time:
20 minutes
Page 1 of 6
PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE
Answer FOUR questions
Question 1.
(a)
. Let
P
the set of relevant price-wealth vectors (
p
,
w
), a subset of
1
L
+
++
ℜ
, and denote by
:
j
xP
→ℜ
±
a
consumer’s Walrasian demand for good
j
,
j
= 1,…
L
, assumed to be strictly positive and differentiable on
P
.
(a).1
. What do we mean when we say that good
j
is a
necessity
for the consumer at (
p
,
w
)? Same
for
luxury
and for
borderline necessity-luxury
.
(a).2
. Show that the concepts of
luxury
and
borderline necessity-luxury
can be characterized by a
property of the budget share function
( , )
j
bpw
of the good. Can you do the same with the concept of
necessity
? Explain
For the rest of this question we consider the indirect utility function
1
()
::
(
,
)
(
)
ln( /
( ))
Fp
vP
vpw
Gp
wCp
−
⎡⎤
=
+
⎢⎥
⎣⎦
,
(1)
where
C
(
p
) >> 0, and the functions
C
,
F
and
G
are such that
v
(
p
,
w
) has the properties of an indirect utility
function on
P
.
(b)
. For
j
= 1, …,
L
, obtain the Walrasian demand function
(,)
j
x pw
±
and the budget share function
j
corresponding to (1).
(c).
Consider first the case of (1) with
G
(
p
) = 0, all
p
. Show that if good
j
is a luxury at some
(,) 0
pw
>>
,
then it is a luxury at
p w
, for all
w
> 0.
(d).
Consider now the general case of (1) where
G
(
p
) is not always zero.
(d).1
. Suppose that good
j
is a luxury at some
>>
. Does it follow that it is a luxury
at
p w
, for all
w
> 0? Explain.
(d).2
. Suppose that all consumers in the economy have identical preferences, of the type
represented by (1).
Under which conditions on the functions
C
,
F
and
G
can the consumers’ aggregate
demand be a function of (only) prices and aggregate wealth? Argue clearly. Discuss the possibility of a
positive representative consumer with these preferences.