3 solve the following two dimensional maximization

3. Solve the following two dimensional maximization problems subject to the linear constraint
w = p
1
x
1
+ p
2
x
2
, p
1
, p
2
> 0. Sketch contour lines of the objective function and the constraint set.
Then compute the change in x
1
with respect to p
2
, and a change in x
1
with respect to w . Assume
α
1
> α
2
> 0. For i and ii , when do the SOSCs hold?
=
p
1
p
2
x
1
x
1
= w

0 − p
1
− p
2
− p
1
α
1
( α
1
− 1) x
α
1
− 2
1
x
α
2
2
α
1
α
2
x
α
1
− 1
1
x
α
2
− 1
2


(1)
So a set of sufficient conditions to ensure this are that 0 < α
1
< 1 and 0 < α
2
< 1, so that the
left-hand side is positive but the right-hand side is negative.
ii. Stone-Geary
f ( x ) = ( x
1
− γ
1
)
α
1
( x
2
− γ
2
)
α
2
The Lagrangian is
L ( x, λ ) = ( x
1
− γ
1
)
α
1
( x
2
− γ
2
)
α
2
− λ ( p
1
x
1
+ p
2
x
2
− w )
The FONCs are
α
1
( x
1
− γ
1
)
α
1
− 1
( x
2
− γ
2
)
α
2
− λp
1
= 0
α
2
( x
1
− γ
1
)
α
1
( x
2
− γ
2
)
α
2
− 1
− λp
2
= 0
− ( p
1
x
1
+ p
2
x
2
− w ) = 0
The first two equations imply
α
1
( x
2
− γ
2
)
α
2
( x
1
− γ
1
)
=
p
1
p
2
4

Solving for x
2
yields
x
2
=
p
1
α
1
p
2
α
2
( x
1
− γ
1
) + γ
2
Substituting this into the constraint yields
p
1
x
1
+
p
1
α
1
α
2
( x
1
− γ
1
) + p
2
γ
2
= w
Solving for x
1
yields
x
∗
1
=
α
1
w − p
2
α
1
γ
2
+ p
1
α
2
γ
1
p
1
( α
1
+ α
2
)
x
∗
2
=
α
2
w − p
1
α
2
γ
1
+ p
2
α
1
γ
2
p
2
( α
2
+ α
1
)
and the comparative statics are
∂x
∗
1
∂w
=
α
1
p
1
( α
1
+ α
2
)
> 0
∂x
∗
1
∂p
2
=
− α
1
γ
2
p
1
( α
1
+ α
2
)
< 0
The bordered Hessian is


0 − p
1