ECO 310 °Fall 2007
Microeconomic Theory °A Mathematical Approach
Problem Set 5 °Answer Key
Question 1:
(a) Each consumer maximizes
U
(
x; y
) =
y
+ 10
x
°
5
x
2
;
subject to the budget constraint
p x
+
y
=
M :
Substituting out
y
, the objective is
F
(
x
)
±
M
°
p x
+ 10
x
°
5
x
2
:
Now
F
0
(
x
) = 10
°
p
°
10
x;
F
00
(
x
) =
°
10
<
0
:
So the FONC yields the global max, and
If
p
²
10
, then
F
0
(0)
³
0
so
x
= 0
is optimum
If
p <
10
, then
F
0
(0)
>
0
and the optimum is
x
= 1
°
p=
10
.
The market demand function just the sum over 160 consumers:
If
p
²
10
, then
X
= 0
If
p <
10
, then
X
= 160
°
16
p
.
(b) Other than the
LRTC
which is as stated, the other costs as functions of
x
are
meaningful only for
x >
0
. The expressions are (5 points):
LRTC
(
x
)
=
°
0
if
x
= 0
4 +
x
2
if
x >
0
LRAC
(
x
)
=
4
=x
+
x
SRTAC
(
x
)
=
1 +
x
2
SRAAC
(
x
)
=
1
=x
+
x
MC
(
x
)
=
2
x
LRAC
0
2
+ 1
,
LRAC
00
3
>
0
. So
LRAC
(
q
)
is minimized at
x
= 2
and the minimum is 4.
SRAAC
0
2
+ 1
,
SRAAC
00
3
>
0
. So
SRAAC
(
q
)
is minimized at
x
= 1
and the minimum is 2.
The ±rm²s short run supply curve coincides with its marginal cost curve so long as the
price does not fall below the minimum
SRAAC
. Thus
p
= 2
x
when
p
²
2
. Or
x
=
°
0
if
p
³
2
p=
2
if
p
²
2
1
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Observe that if
p
= 2
, the ±rm is indi/erent between producing 0 and 1 (it merely loses its
sunk cost, and only just recovers its avoidable cost if it produces 1).
(c) In the long run, with free entry and exit of ±shing ±rms, the industry²s supply curve is
a horizontal line at the level of the minimum
LRAC
, that is, at
p
= 4
. To be more pedantic,
it consists of all points with
p
= 4
and
x
= 2
, 4, 6, 8, 10,
: : :
, but not noting this is not a
problem.
(d) From the supply curve,
p
= 4
. Then from the demand curve,
X
= 16(10
°
4) = 96
.
Each ±rm produces
x
= 2
at the bottom point of its
LRAC
, so there are 48 ±rms. Price
equals the
LRAC
at this point, so each makes zero pro±t. The aggregate consumer surplus
is the area to the left of the market demand curve, so it equals
1
2
(10
°
4) 96 = 288
.
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 Fall '08
 StephenE.Morris
 Supply And Demand, Ann, PY PX XA, PY PX PX

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