University of California, Davis
Date:
August 30, 2007
Department of Agricultural and Resource Economics
Department of Economics
Time:
5 hours
Microeconomics
Reading Time:
20 minutes
PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE
Please answer
four
of the following five parts
Part 1. The present and the future
Let there be two goods: Good 1 is interpreted as
present
consumption
and good 2 as
future
consumption
. We assume that the utility function of the consumer is differentiable, strictly
quasiconcave and locally nonsatiated, and that her Walrasian demand function,
denoted
112
212
((, ,)
, (, ,)
)
xppwxppw
%%
, is welldefined and differentiable, and we restrict our attention
to domains for which
)
is strictly positive.
The consumer is endowed with
ω
1
units of present consumption and
ω
2
units of future
consumption, but has no other sources of wealth (no nonendowment wealth, or profit income). The
market allows the consumer to exchange the two goods, in any direction and at any scale, at the
constant rate of 1 +
r
units of future consumption for each unit of present consumption. The real
number
r
is called the
market interest rate
, which the consumer takes as given. If, given
r
, the
consumer chooses
x
1
<
ω
1
(resp.
x
1
>
ω
1
, resp.
x
1
=
ω
1
), then we say that she is a
saver
(resp. a
borrower
, resp.
autarkic
) at
r
. We refer to the difference
s
≡
ω
1

x
1
as the
savings
of the consumer,
which can be positive, negative or zero.
1.1.
Write the consumer optimization problem in terms of the parameters
p
1
and
p
2
,
interpreted as the prices of present and future consumption, respectively. Assume that the solution
function, denoted
112 212
ˆˆ
((,)
,(,)
)
xpp xpp
, is well defined and differentiable.
What is the
relationship between
p
1
,
p
2
and
r
?
Graphically represent the consumer optimization problem in
consumption space.
1
.
2
. Write the Slutsky equation for
12
ˆ
(, )
,,
1
,
2
k
j
xpp
kj
p
∂
=
∂
.