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Winter 2011
SOLUTIONS FOR PROBLEM SET I
1.a. We need
a
and
b
to be positive for a downward sloping demand curve and
k
to be positive for
positive costs.
1.b. The monopolist profit is
2
p
xk
x t
x
where
p
and
x
satisfy the demand function.
Solve the
demand function for
p
in terms of
x
to get
ax
p
bb
, and substitute this into the expression for
profit to get
2
2
ax
x
kx
tx
.
The first order condition for choosing
x
to maximize profit is
2
20
dax
kx t
dx
b
b
and the (sufficient) second order condition is
2
2
2
d
k
dx
b
which
is satisfied since
b
and
k
are positive.
Solve the first order condition to get the profitmaximizing
x
in terms of
t
:
*
()
x xt
where
*
2(1
)
ab
t
xt
bk
.
An increase in
t
reduces the profitmaximizing
output because
*
0
)
dx t
b
dt
bk
since
b
and
k
are both positive.
1.c. The (sufficient) second order condition for profit maximization requires that
1
k
b
which
is weaker than the conditions we imposed in part a above.
1.d. Multiply through the first order condition from part b by
b
and rearrange to get
)
t
b
kx
.
Substitute
*
)
t
bk
from part b for
x
to see that
)
)
t
t
b
k
t
bk
which holds identically for all values of
t
.
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 Spring '08
 Ellis,G

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