The document discusses how demand changes in response to price changes using different utility functions as examples. When the price of a good increases, holding other prices and income constant, demand for that good will decrease along its ordinary demand curve. The curve tracing out the combinations of the good and other goods consumed at different price levels is the price offer curve. For a Cobb-Douglas utility function, the price offer curve is flat and the demand curve is a rectangular hyperbola. For perfect complements, demand decreases as price increases along a linear curve. For perfect substitutes, demand is either zero or the full income is spent on the good with the lower price. The inverse demand curve shows the price corresponding to a given quantity demanded
2. Properties of Demand Functions
Comparative statics analysis of
ordinary demand functions -- the
study of how ordinary demands
x1*(p1,p2,y) and x2*(p1,p2,y) change as
prices p1, p2 and income y change.
3. Own-Price Changes
How does x1*(p1,p2,y) change as p1
changes, holding p2 and y constant?
Suppose only p1 increases, from p1’
to p1’’ and then to p1’’’.
15. Own-Price Changes
x2
Fixed p2 and y.
p1
p1’’’
p1’’
p1’
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1
x1*
16. Own-Price Changes
x2
Fixed p2 and y.
p1
Ordinary
demand curve
for commodity 1
p1’’’
p1’’
p1’
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1
x1*
17. Own-Price Changes
x2
Fixed p2 and y.
p1
Ordinary
demand curve
for commodity 1
p1’’’
p1’’
p1’
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1
x1*
18. Own-Price Changes
x2
Fixed p2 and y.
p1 price
offer
curve
p1
Ordinary
demand curve
for commodity 1
p1’’’
p1’’
p1’
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1
x1*
19. Own-Price Changes
The curve containing all the utilitymaximizing bundles traced out as p 1
changes, with p2 and y constant, is
the p1- price offer curve.
The plot of the x1-coordinate of the p1price offer curve against p1 is the
ordinary demand curve for
commodity 1.
21. Own-Price Changes
What does a p1 price-offer curve look
like for Cobb-Douglas preferences?
Take
a b
U( x1 , x 2 ) = x1 x 2 .
Then the ordinary demand functions
for commodities 1 and 2 are
22. Own-Price Changes
a
y
×
a + b p1
and
b
y
*
x 2 (p1 , p 2 , y) =
×
.
a + b p2
*
x1 (p1 , p 2 , y) =
Notice that x2* does not vary with p1 so the
p1 price offer curve is
23. Own-Price Changes
a
y
×
a + b p1
and
b
y
*
x 2 (p1 , p 2 , y) =
×
.
a + b p2
*
x1 (p1 , p 2 , y) =
Notice that x2* does not vary with p1 so the
p1 price offer curve is flat
24. Own-Price Changes
a
y
×
a + b p1
and
b
y
*
x 2 (p1 , p 2 , y) =
×
.
a + b p2
*
x1 (p1 , p 2 , y) =
Notice that x2* does not vary with p1 so the
p1 price offer curve is flat and the ordinary
demand curve for commodity 1 is a
25. Own-Price Changes
a
y
×
a + b p1
and
b
y
*
x 2 (p1 , p 2 , y) =
×
.
a + b p2
*
x1 (p1 , p 2 , y) =
Notice that x2* does not vary with p1 so the
p1 price offer curve is flat and the ordinary
demand curve for commodity 1 is a
rectangular hyperbola.
29. Own-Price Changes
What does a p1 price-offer curve look
like for a perfect-complements utility
function?
U( x1 , x 2 ) = min{ x1 , x 2} .
Then the ordinary demand functions
for commodities 1 and 2 are
31. Own-Price Changes
*
*
x1 (p1 , p 2 , y) = x 2 (p1 , p 2 , y) =
y
.
p1 + p 2
With p2 and y fixed, higher p1 causes
smaller x1* and x2*.
32. Own-Price Changes
*
*
x1 (p1 , p 2 , y) = x 2 (p1 , p 2 , y) =
y
.
p1 + p 2
With p2 and y fixed, higher p1 causes
smaller x1* and x2*.
As
p1 → 0,
y
*
*
x1 = x 2 →
.
p2
33. Own-Price Changes
*
*
x1 (p1 , p 2 , y) = x 2 (p1 , p 2 , y) =
y
.
p1 + p 2
With p2 and y fixed, higher p1 causes
smaller x1* and x2*.
y
*
*
.
As p1 → 0, x1 = x 2 →
p2
*
*
As p1 → ∞ , x1 = x 2 → 0.
35. Own-Price Changes
p1
Fixed p2 and y.
x2
p1 = p1’
y/p2
x* =
2
p1’
y
p1’+ p 2
x* =
1
x* =
1
y
p1’+ p 2
x1
y
p1 ’+ p 2
x1*
36. Own-Price Changes
p1
Fixed p2 and y.
x2
p1 = p1’’
y/p2
p1’’
p1’
x* =
2
y
p1’’+ p 2
x* =
1
x* =
1
y
p1’’+ p 2
x1
y
p1’’+ p 2
x1*
37. Own-Price Changes
Fixed p2 and y.
x2
p1 = p1’’’
y/p2
p1
p1’’’
p1’’
p1’
x* =
2
x* =
1
y
p1’’’ + p 2
x* =
1
y
p1’’’ + p 2
y
p1’’’ + p 2
x1
x1*
38. Own-Price Changes
Fixed p2 and y.
p1
Ordinary
demand curve
for commodity 1
is
y
*
x1 =
.
p1 + p 2
p1’’’
x2
p1’’
y/p2
x* =
2
p1’
y
p1 + p 2
y
p2
x* =
1
y
p1 + p 2
x1
x1*
39. Own-Price Changes
What does a p1 price-offer curve look
like for a perfect-substitutes utility
function?
U( x1 , x 2 ) = x1 + x 2 .
Then the ordinary demand functions
for commodities 1 and 2 are
40. Own-Price Changes
0
*
x1 (p1 , p 2 , y) =
, if p1 > p 2
0
*
x 2 (p1 , p 2 , y) =
, if p1 < p 2
and
y / p1 , if p1 < p 2
y / p 2 , if p1 > p 2 .
48. Own-Price Changes
Fixed p2 and y.
p1
Ordinary
demand curve
for commodity 1
p1’’’
y
*
x1 =
p1
x2
p2 = p1’’
y
p2
p1 price
offer
curve
p1’
y
*
0 ≤ x1 ≤
p2
x1
x1*
49. Own-Price Changes
Usually we ask “Given the price for
commodity 1 what is the quantity
demanded of commodity 1?”
But we could also ask the inverse
question “At what price for
commodity 1 would a given quantity
of commodity 1 be demanded?”
52. Own-Price Changes
p1
Given p1’, what quantity is
demanded of commodity 1?
Answer: x1’ units.
The inverse question is:
Given x1’ units are
demanded, what is the
price of
x1* commodity 1?
x1’
53. Own-Price Changes
p1
p1’
Given p1’, what quantity is
demanded of commodity 1?
Answer: x1’ units.
The inverse question is:
Given x1’ units are
demanded, what is the
price of
x1* commodity 1?
x1’
Answer: p1’
54. Own-Price Changes
Taking quantity demanded as given
and then asking what must be price
describes the inverse demand
function of a commodity.
55. Own-Price Changes
A Cobb-Douglas example:
ay
*
x1 =
( a + b )p1
is the ordinary demand function and
p1 =
ay
*
( a + b ) x1
is the inverse demand function.
64. Income Changes
x2
Fixed p1 and p2.
y’ < y’’ < y’’’
Income
offer curve
x2’’’
x2’’
x2 ’
y
y’’’
y’’
y’
x1’ x1’’’
x1’’
x1
x1’ x1’’’
x1’’
x1*
65. Income Changes
x2
Fixed p1 and p2.
y’ < y’’ < y’’’
Income
offer curve
x2’’’
x2’’
x2 ’
y
y’’’
y’’
y’
x1’ x1’’’
x1’’
x1
Engel
curve;
good 1
x1’ x1’’’ x1*
x1’’
66. Income Changes
x2
y
Fixed p1 and p2.
y’’’
y’’
y’ < y’’ < y’’’
y’
Income
offer curve
x2’’’
x2’’
x2 ’
x1’ x1’’’
x1’’
x1
x2’ x2’’’
x2’’
x2*
67. Income Changes
x2
y
Fixed p1 and p2.
y’’’
y’’
y’ < y’’ < y’’’
y’
Income
offer curve
x2’’’
x2’’
x2 ’
x1’ x1’’’
x1’’
x1
Engel
curve;
good 2
x2’ x2’’’
x2’’
x2*
68. Income Changes
x2
y
Fixed p1 and p2.
y’’’
y’’
y’ < y’’ < y’’’
y’
Income
offer curve y
x2’’’
x2’’
x2 ’
y’’’
y’’
y’
x1’ x1’’’
x1’’
x1
Engel
curve;
good 2
x2’ x2’’’
x2’’
x2*
Engel
curve;
good 1
x1’ x1’’’ x1*
x1’’
69. Income Changes and CobbDouglas Preferences
An example of computing the
equations of Engel curves; the CobbDouglas case.
a b
U( x1 , x 2 ) = x1 x 2 .
The ordinary demand equations are
*
x1 =
ay
by
*
; x2 =
.
( a + b )p1
( a + b )p 2
70. Income Changes and CobbDouglas Preferences
*
x1 =
ay
by
*
; x2 =
.
( a + b )p1
( a + b )p 2
Rearranged to isolate y, these are:
( a + b )p1 *
y=
x1 Engel curve for good 1
a
( a + b )p 2 *
y=
x 2 Engel curve for good 2
b
71. Income Changes and CobbDouglas Preferences
y
y
( a + b )p1 *
y=
x1
a
Engel curve
for good 1
x1 *
( a + b )p 2 *
y=
x2
b
x2*
Engel curve
for good 2
72. Income Changes and PerfectlyComplementary Preferences
Another example of computing the
equations of Engel curves; the
perfectly-complementary case.
U( x1 , x 2 ) = min{ x1 , x 2} .
The ordinary demand equations are
*
*
x1 = x 2 =
y
.
p1 + p 2
73. Income Changes and PerfectlyComplementary Preferences
*
*
x1 = x 2 =
y
.
p1 + p 2
Rearranged to isolate y, these are:
*
y = (p1 + p 2 )x1
*
y = (p1 + p 2 )x 2
Engel curve for good 1
Engel curve for good 2
78. Income Changes
Fixed p1 and p2.
x2
y’ < y’’ < y’’’
y
x2’’’
x2’’
x2’
y’’’
y’’
y’
x1’ x1’’’
x1’’
x1
Engel
curve;
good 1
x1’ x1’’’
x1*
x1’’
79. Income Changes
y
Fixed p1 and p2.
x2
y’’’
y’’
y’ < y’’ < y’’’
y’
Engel
curve;
good 2
x2’ x2’’’
x2’’
x2’’’
x2’’
x2’
x1’ x1’’’
x1’’
x1
x2*
80. Income Changes
y
Fixed p1 and p2.
x2
y’’’
y’’
y’ < y’’ < y’’’
y’
y
x2’’’
x2’’
x2’
y’’’
y’’
y’
x1’ x1’’’
x1’’
x1
Engel
curve;
good 2
x2’ x2’’’
x2’’
x2*
Engel
curve;
good 1
x1’ x1’’’
x1*
x1’’
81. Income Changes
Fixed p1 and p2.
*
y = (p1 + p 2 )x 2
y
y’’’
y’’
y’
y
*
y = (p1 + p 2 )x1
y’’’
y’’
y’
Engel
curve;
good 2
x2’ x2’’’
x2’’
x2*
Engel
curve;
good 1
x1’ x1’’’
x1*
x1’’
82. Income Changes and PerfectlySubstitutable Preferences
Another example of computing the
equations of Engel curves; the
perfectly-substitution case.
U( x1 , x 2 ) = x1 + x 2 .
The ordinary demand equations are
83. Income Changes and PerfectlySubstitutable Preferences
0
*
x1 (p1 , p 2 , y) =
, if p1 > p 2
y / p1 , if p1 < p 2
, if p1 < p 2
0
*
x 2 (p1 , p 2 , y) =
y / p 2 , if p1 > p 2 .
84. Income Changes and PerfectlySubstitutable Preferences
0
*
x1 (p1 , p 2 , y) =
, if p1 > p 2
y / p1 , if p1 < p 2
, if p1 < p 2
0
*
x 2 (p1 , p 2 , y) =
y / p 2 , if p1 > p 2 .
Suppose p1 < p2. Then
85. Income Changes and PerfectlySubstitutable Preferences
0
*
x1 (p1 , p 2 , y) =
, if p1 > p 2
y / p1 , if p1 < p 2
, if p1 < p 2
0
*
x 2 (p1 , p 2 , y) =
y / p 2 , if p1 > p 2 .
y
*
*
Suppose p1 < p2. Then x1 =
and x 2 = 0
p1
86. Income Changes and PerfectlySubstitutable Preferences
0
*
x1 (p1 , p 2 , y) =
, if p1 > p 2
y / p1 , if p1 < p 2
, if p1 < p 2
0
*
x 2 (p1 , p 2 , y) =
y / p 2 , if p1 > p 2 .
y
*
*
Suppose p1 < p2. Then x1 =
and x 2 = 0
p1
*
*
y = p1x1 and x 2 = 0.
87. Income Changes and PerfectlySubstitutable Preferences
y
y
*
y = p1x1
x1*
Engel curve
for good 1
x* = 0.
2
x2 *
0
Engel curve
for good 2
88. Income Changes
In every example so far the Engel
curves have all been straight lines?
Q: Is this true in general?
A: No. Engel curves are straight
lines if the consumer’s preferences
are homothetic.
89. Homotheticity
A consumer’s preferences are
homothetic if and only if
(x1,x2) (y1,y2) ⇔ (kx1,kx2)
(ky1,ky2)
for every k > 0.
That is, the consumer’s MRS is the
same anywhere on a straight line
drawn from the origin.
90. Income Effects -- A
Nonhomothetic Example
Quasilinear preferences are not
homothetic.
U( x1 , x 2 ) = f ( x1 ) + x 2 .
For example,
U( x1 , x 2 ) = x1 + x 2 .
96. Income Effects
A good for which quantity demanded
rises with income is called normal.
Therefore a normal good’s Engel
curve is positively sloped.
97. Income Effects
A good for which quantity demanded
falls as income increases is called
income inferior.
Therefore an income inferior good’s
Engel curve is negatively sloped.
98. Income Changes; Goods
y
1 & 2 Normal
x2
Income
offer curve
x2’’’
x2’’
x2 ’
y’’’
y’’
y’
y
y’’’
y’’
y’
x1’ x1’’’
x1’’
x1
Engel
curve;
good 2
x2’ x2’’’
x2’’
x2*
Engel
curve;
good 1
x1’ x1’’’ x1*
x1’’
110. Ordinary Goods
Fixed p2 and y.
Downward-sloping
p1 demand curve
x2
⇔
p1 price
offer
curve
Good 1 is
ordinary
x1*
x1
111. Giffen Goods
If, for some values of its own price,
the quantity demanded of a good
rises as its own-price increases then
the good is called Giffen.
114. Ordinary Goods
Demand curve has
a positively
p1
sloped part
Fixed p2 and y.
x2
⇔
p1 price offer
curve
Good 1 is
Giffen
x1*
x1
115. Cross-Price Effects
If an increase in p2
– increases demand for commodity 1
then commodity 1 is a gross
substitute for commodity 2.
– reduces demand for commodity 1
then commodity 1 is a gross
complement for commodity 2.
116. Cross-Price Effects
A perfect-complements example:
y
*
x1 =
p1 + p 2
so
*
∂ x1
y
=−
< 0.
2
∂ p2
(p +p )
1
2
Therefore commodity 2 is a gross
complement for commodity 1.
120. Cross-Price Effects
A Cobb- Douglas example:
by
*
x2 =
( a + b )p 2
so
*
∂ x2
= 0.
∂ p1
Therefore commodity 1 is neither a gross
complement nor a gross substitute for
commodity 2.
