University of California, Davis
Date: August 31, 2006
Department of Economics
Time:
4 hours
Microeconomics
Reading Time:
20 minutes
PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE
==================================================================================
Please answer four
four
four of the following equally weighted five questions
1.
Variations on variable varieties
We consider a model of consumer preferences where the different commodities are
viewed as different varieties of a good, and the number
L
of varieties is variable. Variations of
this model have been extensively used in industrial organization and international trade.
Formally, we postulate that the preferences of our consumer are represented by the
following family of utility functions, indexed by the number of varieties present in the market:
)
,
1
(
)
1
,
0
(
)
0
,
(
,
,
)
,...,
(
1
1
1
1
∞
∪
∪
∞
∈
σ
ℜ
∈
γ
=

σ
σ
=
σ

σ
γ
∑
L
i
i
L
L
x
L
x
x
u
.
(1)
In words,
)
,...,
(
1
L
L
x
x
u
is the utility that our consumer gets when, in a
L
variety world, she
consumes
x
j
units of variety
j
,
j
= 1,…,
L
.
Our consumer has wealth
w
> 0, takes prices (
p
1
, …,
p
L
) as given, and is always able to
satisfy her Walrasian demand.
(a).
Let
γ
= 0.
We consider different worlds characterized by different numbers of
varieties, but we assume that
p
1
=
p
2
= … =
p
L
=
p
, for every
L
, i. e., every price is always
equal to a given positive number
p
, or, in other words, prices are equal across worlds (or
numbers of varieties) and across varieties.
Does a higher
L
benefit the consumer? Argue clearly, separately considering the
following three cases.
Case 1
:
σ
> 1 (this is the standard case);
Case 2
:
σ
∈
(0,1);
Case 3
:
σ
< 0.
(b).
Now
γ
is unrestricted, but we let
σ
> 1 (as in Case 1 of part (a)). Compute the own
price elasticity of demand for variety
j
(
j
= 1,…,
L
) for the utility function given in (1). (Hint
.
Recall that the Walrasian demand function for good
j
for the CES function
1
1
1

σ
σ
=
σ

σ
∑
L
i
i
x
,