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Implicit function theorem question: ARE prelim, 2008
Question: A monopolist sells in two countries: 1 and 2. It
produces a good at a
constant marginal cost ofcand cannot produce more than a total
of Qunits.
Country 1 imposes a per unit tax ofunits on the good sold in
that country.
Assume that the constraint binds.
1 For a given value of , show how the equilibrium in the two
countries are
determined.
2 Show how these equilibrium prices change as increases.
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Answer to ARE prelim, 2008 question
Answer: Assume that inverse demand curves are concave in
price.
The monopolists optimization problem is
maxp1,p2
(p1 )D1(p1) +p2D2(p2)c(D1(p1) +D2(p2))
s.t. D1(p1) +D2(p2)Q
The Lagrangian is
L(p1,p2,;) = (p1 )D1(p1) +p2D2(p2)c(D1(p1) +D2(p2))
+ (QD1(p1)D2(p2))
Assuming capacity constraint binds, the first order conditions
are
Lp1 = D1(p1) + (p1 c)D1(p1) = 0
Lp2 = D2(p2) + (p2 c)D2(p2) = 0
L = QD1(p1)D2(p2) = 0
Solution is a triple(p
1,p
2,
)which solves this system,given.() August 18, 2013 2 / 5
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Q
Marginal revenue
(p2)
MR2()
c
MR1(, )
MR1(, )
c+ ()
c+ ()
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Lp1 = D1(p1) + (p1 c)D1(p1) = 0
Lp2 = D2(p2) + (p2 c)D2(p2) = 0
L = QD1(p1)D2(p2) = 0
The Hessian of the Lagrangian w.r.t. endog vars is:
HLp, =
2D
1(p
1 ) + (p
1c
)D1 (p1 ) 0 D
1(p
1 )
0 2D2(p2 ) + (p
2c
)D2 (p2 ) D
2(p
2 )
D1(p1 ) D
2(p
2 ) 0
HL =
D1(p1)0
0
.
Hence from the implicit function theorem, we have
dp1ddp2dd
d
= HL1(p,)
D1(p1)
0
0
= HL1(p,)
D1(p1)
0
0
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Its straightforward to check that the determinant ofHL(p,)
is
D2(p
2)
2
2D1(p1) + (p
1 c
)D1 (p1)
+D
1(p
1)2
2D
2(p
2) + (p
2 c
)D
2 (p
2)
>0
Applying Cramers rule, we have
1
= det
D1(p
1 ) 0 D
1(p
1 )
0 2D2(p2 ) + (p
2 c
)D2 (p2 ) D
2(p
2 )
0 D2(p2 ) 0
det(HL(p,))
= D1(p1)D2(p2)2
det(HL(p,))> 0
2
= det
2D1(p
1 ) + (p
1 c
)D1 (p1) D
1(p
1 ) D
1(p
1 )
0 0 D2(p1 )
D1(p2 ) 0 0
det(HL(p,))
= D2(p1)D
1(p
2)
2det(HL(p,))< 0
= det
2D1(p
1 ) + (p
1 c
)D1 (p1) 0 D
1(p
1 )
0 2D2(p2 ) + (p
2 c
)D2 (p2) 0
D1(p1 ) D
2(p
2 ) 0
det(
=
2D2(p2) + (p
2 c
)D2 (p2)D1(p
2)
2
det(HL(p,))< 0
() August 18, 2013 5 / 5
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