 
    

  
University
of
Wisconsin
Economics 301: Intermediate Microeconomic Theory
Korinna K. Hansen
Answers to Practice Problems
(part
2)
1).
a). False. The utility function gives us information about risk aversion and not the price
of
t1I'1
insurance. Risk averse individuals exhibit diminishing marginal utility.
TU
Tu
1L{
b). True. To derive the aggregate demand for a
p
private good,
we
need to add up the two demand
curves horizontally.
We
add up the quantities at
each price. x =
Xl
+
X2
= 1,000
2p
+500 
p.
Therefore X= 1,500 
3p for p<500.
500
I
~
p=
500
=
.3
r
ISOO
X

c). True. With the reduction in the barley crop, the supply will decrease and cause the price
of
barley to increase. At the higher price less will be exchanged. Since the demand
is
elastic the
decrease
in
quantity will be larger than the increase
in
price, and total revenues will therefore
fall.
(Remember
dR/dq
= p
(1
lie)).
d). True.
We
can use the equation for the elasticity
of
demand here e= (dq/dp
)(p/q)
= (1/slope)
(p/q)
and calculate the elasticity at the two prices
10
and 20 cents. The slope
of
this linear
demand curve
is
 1/10. When the price
is
10
cents, x = 999 and when the price
is
20 cents, x =
998. Then the elasticity at p =
10
cents: e = (10)
(0.1/999)
= 0.001 and the elasticity at p=20
cents: e = (10)
(0.2/998)
= 0.002. So as the price
of
potatoes changes from
10
to 20 cents the
price elasticity
of
demand increases in absolute value.
e). True. The elasticity
of
demand
is
e=
(dq/d~)(p/q),
and q =
2/p.
Therefore, dq/dp =
2(_1)p2
and
p/q
=
p/(2/p)
=
p2
12
. Then e= 2(_l)p2) (p
12)
=1
f).
True. To add up the two demand curves horizontally
we
first need to solve them for the
quantities and
we
have
Xl
=
15
 P and
X2
=
20/3 1/3
p.
Then the aggregate demand will be
X=Xl+X2
=
15
 P +
20/3 1/3
P =
65/3

413
p.
So, X=
(65/3)

(413)
P and
if
we
put
p=ll
in
the aggregate demand equation we find that x =
65/3
(4/3)
(11) =21/3 =7
g). True. The consumers will pay more than half the
P
tax
if
their demand
is
relatively more inelastic than
l'
P,'t
the producer supply. Similarly, the producers will
~~rt'v
pay more than half the tax
if
their supply
is
relatively
more inelastic than the consumers demand. The party
Pl.
with the most inelastic curve pays most
of
the tax.
P.3
tt
P
!'OJ
~
C
e.r
fh'l'(.f
....
~o?
£Q.)(
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5
f
h).
False.
If
the amount
of
the good supplied
is
independent
R
=(1
of
the price, then the supply curve
is
perfectly inelastic.
B.
I
Initial equilibrium quantity
and
price are at
ql
and
Pl.
Then
after tax consumers
will
still pay
PI
=
P2,
but producers
receive the lower price
P3,
and
not
PI
anymore.
The
difference
PIP3
= quantity
tax.
&1,
~
i)
False.
If
you double both inputs
f(2XI,
2X2)=
min
(4XI+2X2,
2XI+4x2)
= 2
min
(2XI+X2,
xI+2x2).
So,
this technology exhibits constant returns to
scale.
j).
True.
The
technical rate
of
substitution,
TRS
=
MPIIMP2
l12
l12
MPI
= 2
(2
XI+
4X2r
(112)
and
MP2
= 4
(2
XI+
4X2r
(112).
Therefore
TRS
=
1;2
k).
False. This
firm
could experience constant, increasing or decreasing returns to scale
depending on the specific functional
form
of
the production function
f(XI,
X2)
=
axl+bx2.
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 Spring '08
 Hansen
 Microeconomics, Supply And Demand, Utility, demand curves, linear demand curve, certainty equivalent income

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