5/18/2010
ProblemSetsSpring2010.doc
2:44 PM
1
Robert Deacon
Econ. 210C
Spring 2010
Problem Set #1
In the following problems you may assume that the underlying preferences make it
legitimate to use partial equilibrium analysis.
1.
Market demand for a particular good is given by the non-increasing function
)
(
c
p
X
while market supply is given by the non-decreasing function
)
(
s
p
Q
where
p
c
and
p
s
indicate the prices paid by consumers and the price received by suppliers,
respectively. A tax of
t
per unit is imposed on consumers’ purchases.
a.
Derive an expression that characterizes the effect of the tax on the price producers
receive, i.e.,
dt
t
dp
s
/
)
(
, in terms of the price elasticities of supply and demand.
b.
What does your expression indicate about the effect of the tax on producers’
prices in the following special cases: (i) demand is perfectly price elastic, (ii)
demand is perfectly price inelastic, (iii) elasticities of supply and demand are non-
zero, finite and equal in absolute value. Illustrate these cases with a diagram.
2.
A perfectly competitive industry is
composed of numerous firms with identical cost
functions. In order to produce any positive output a firm must use a fixed amount of
one input, e.g., entrepreneurship, but all other inputs are variable. As a result each
firm’s cost function is given by:
2
9
)
(
j
j
q
q
c
+
=
for
0
>
j
q
, and
0
)
(
=
j
q
c
for
0
=
j
q
, where
q
j
is
j
th
firm’s output.
Aggregate demand (in inverse form) is given by
q
P
1
.
36
−
=
, where
q
is the total
quantity consumed.
a.
What is the long run (allowing for entry and exit) equilibrium price, output per
firm and number of firms?
b.
Suppose a tax of 6 per unit is imposed on all units produced. Calculate the
equilibrium price and output in a short-run situation where the number of firms
remains fixed but the fixed cost cannot be avoided by shutting down. Calculate
the new long run equilibrium price, output and number of firms.
3.
Two consumers,
2
,
1
=
i
, have preferences given by
i
i
i
i
m
a
x
U
+
+
=
)
ln(
where
3
1
1
=
a
,
3
2
2
=
a
and
0
≥
i
x
,0
≥
i
m
are
i
’s consumption of an ordinary good and a
numeraire. Consumer
i
is endowed with
ω
i
units of numeraire. The price of the
ordinary (non-numeraire) good is denoted
p
and its supply function is given by
q
=
p
,
where
q
is aggregate supply. The market is competitive. (You may consider the two
consumers as representing large numbers of consumers of two types.)
a.
Find the equilibrium price, total output, and consumption by each consumer.
b.
Suppose a regulatory agency imposes a price ceiling
3
1
≤
p
. The supply function
remains unchanged and the regulator allocates the resulting quantity supplied among
the two consumers in a way that maximizes their aggregate utility. Is consumer 2
better or worse off as a result? How about consumer 1? (It is okay to answer by
setting up the utility comparison but without carrying out the calculation.) Explain.