11/16/2011
260AHomeworkProblems
12:53 PM
1
Econ 260A
Homework problems
1.
Consider an economy populated by a representative consumer with utility function
m
R
F
U
2
1
2
1
, where
F
,
R
and
m
are consumption of food, recreation and a numeraire.
F
and
R
use water as the only input and are produced according to the production
functions
F
W
F
and
R
W
R
, and the endowments of water and numeraire
are
1
R
F
W
W
and
respectively.
a.
Find the competitive equilibrium and consumer utility in a situation where there are
markets for
m
and
W
F
but not for
W
R
and use
F
has the right to use the entire stock.
That is, our representative consumer can use as much of the water as she wishes for
F
, up to the entire endowment.
b.
Find the representative consumer’s utility with complete markets.
2.
A representative consumer’s utility function is
b
a
w
m
bw
aw
U
2
for
2
and
m
b
a
U
4
2
otherwise, where
w
and
m
are consumption of groundwater and a numeraire.
The consumer is endowed with
units of numeraire. Extracting groundwater uses numeraire
according to the cost function
cDw
C
, where
D
is the depth of the aquifer. Extraction
changes the depth according to
R
w
D
. The variables
w
,
m
,
D
are all functions of time;
parameters
a
,
b
,
c
,
R
,
, and
r
(the interest rate) are all positive.
a.
Find the steady state value of
D
that would result from maximizing the present value
of utility.
b.
Find the steady state value of
D
that would result from independent extraction, where
the representative consumer
takes
D
as given, independent of her own actions. How
would steady state utility be affected by a technological advance that cuts
c
in half?
c.
Imagine that a management agency attempts to implement the solution in part a. by
creating ‘rights’ to water in the aquifer. Each user would receive a ‘water account’
with a pro rata share of water then in storage; a pro rata share of recharge would be
credited (added) to each account and the user’s withdrawal would be deducted. Will
this policy achieve the efficient outcome?
3.
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 Fall '10
 DEACON
 Economics, Steady State, Utility, representative, representative consumer, competitive equilibrium, nonrenewable resource

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