Professor Jay Bhattacharya
Spring 2001
Econ 11Lecture 6
1
Spring 2001
Econ 11Lecture 6
1
Demand II
• Recap: last lecture we covered:
– Income Expansion Paths and Engel curves
– Inferior and Normal Goods
– Necessities and Luxuries
– “Marshallian” Demand Curves
Spring 2001
Econ 11Lecture 6
2
Example: Calculating IEPs and
Engel Curves
•
Find the IEP and Engel Curve for a consumer
with
• To find the solution:
– Solve for the Marshallian demand curves. This will
automatically give you the Engel Curve
– Solve each demand curve for income
– Set these equations equal to each other to derive the
IEP.
()
α
−
=
1
,
:
Function
Utility
Douglass
Cobb
y
x
y
x
U
I
y
p
x
p
y
=
+
x
:
Constraint
Budget
Spring 2001
Econ 11Lecture 6
3
Solved Example
• Set up the Lagrangian:
• Calculate the first order conditions:
I
y
p
x
p
y
x
L
y
x
−
+
−
=
−
λ
1
0
=
−
ú
û
ù
ê
ë
é
=
∂
∂
y
p
y
x
y
L
0
1
=
−
ú
û
ù
ê
ë
é
=
∂
∂
−
x
p
x
y
x
L
0
=
−
+
=
∂
∂
I
y
p
x
p
L
y
x
Spring 2001
Econ 11Lecture 6
4
Solved Example (II)
• FindtheMarshalliandemandcurves:
• These demand curves are the same as the Engel
curves, since they show how the optimal levels of
x
and
y
change with income.
• Note that for CobbDouglass utility, Engel curves
are linear in income.
x
y
x
p
I
I
p
p
x
=
,
,
y
y
x
p
I
I
p
p
y
−
=
1
,
,
Spring 2001
Econ 11Lecture 6
5
Solved Example (III)
• Solve each demand curve for income:
• Setting these equations equal to each other
gives the income expansion path:
• For CobbDouglass utility, the IEP is linear.
*
x
p
I
x
=
−
=
1
*
y
p
I
y
y
x
x
y
p
x
p
y
x
p
y
p
*
*
*
*
1
1
−
=
Þ
=
−
Spring 2001
Econ 11Lecture 6
6
What happens to demand when price
changes?
x
2
x
1
2
*
1
p
p
slope
−
=
2
0
1
p
p
slope
−
=
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View Full DocumentProfessor Jay Bhattacharya
Spring 2001
Econ 11Lecture 6
2
Spring 2001
Econ 11Lecture 6
7
x
1
0
1
p
*
1
p
Marshallian Demand Curve
p
1
Spring 2001
Econ 11Lecture 6
8
What Causes the Change in
Demand?
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 Fall '08
 Woroch
 Microeconomics, demand curves, Professor Jay Bhattacharya, Econ 11Lecture

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