Unformatted text preview: Solutions to Excercise in CH3 Slides
Bin Xie
2/10/2015
Example on Page 13: QuasiLinear Utility Function U = u(q1 ) + q2 . du
U1
du
Solution: U1 =
, U2 = 1. M RS = −
=−
.
dq1
U2
dq1
ρ ρ Excercise (1) on Page 26: U = (q1 + q2 )1/ρ ∗ ∗
d(q2 /q1 ) d(p2 /p1 )
substitution
is a
∗ ∗ / p /p
q2 /q1
2
1
1 ρ
ρ 1−ρ
Solution: U1 = (q1 + q2 ) ρ ρ
ρ−1
q2 .
U1
q1
M RS = −
= −( )ρ−1 .
U2
q2 ρ
ρ
(q1 + q2 ) Show the M RT and show that the elasticity of constant. ρ−1
ρ
ρ
· ρq1
= (q1 + q2 ) 1−ρ
ρ ρ−1
q1 ; U2 = 1 ρ
ρ−1
ρ 1−ρ
(q1 + q2 ) ρ · ρq2
=
ρ 1−ρ
ρ M RS =
∗
q1 ρ−1
p1
M RT , i.e. −( ∗ )
=− .
q2
p2
∗
p1 1/(ρ−1)
q1
Rewrite the equation:
. The elasticity of substitution could also be rewritten
∗ = (p )
q2
2
∗ ∗
∗ ∗
∗ ∗
d(q2 /q1 ) q2 /q1
d(q2 /q1 ) d(p2 /p1 )
=
/
.
as
∗ /q ∗ / p /p
q2 1
d(p2 /p1 ) p2 /p1
2
1
∗ ∗
(Think q2 /q1 and p2 /p1 as two single variables instead of four variables. You can
As we know, solving the utility maximization/expenditure minimization problem yields: ∗ ∗
regard q2 /q1 as q and p2 /p1 as p. So the equation becomes q = p1/(ρ−1) and the elasticity becomes
So the dq p
. Now it becomes like a problem to derive an elasticity of demand.)
dp q
1
dq q
1
p
1
elasticity of substitution is
=
p ρ−1 −1 1/(ρ−1) =
which is a constant.
dp p
ρ−1
ρ−1
p The rest two excercise questions on page 26 are two examples of a CobbDouglas utility function.
You can do them by yourself and check the answers based on the notes you got from the lecture. 1...
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Full Document
 Fall '10
 Raven
 Supply And Demand, Trigraph, Bin Xie

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