Chapter Six

1
Chapter Six

• Demand

2
Demand

• Last Chapter Derived optimal demands/choices
  under different types of preferences.
• x1(p1,p2,m)
• x2(p1,p2,m)
• This Chapter Look at properties of the demand
  functions. See how demand changes when prices or
  income change.

3
Demand Overview

• Income Changes
• Income offer curves (income expansion paths)
• Engel curves
• Normal versus Inferior Goods
• Own Price Changes
• Price offer curves
• Demand curves
• Ordinary versus Giffen Goods
• Cross Price Changes
• Gross Complements
• Gross Substitutes

4
Properties of Demand Functions

• Comparative statics analysis of ordinary demand
  functions -- the study of how ordinary demands
  x1(p1,p2,m) and x2(p1,p2,m) change as prices
  p1, p2 and income m change.

5
Income Offer and Engel Curves

• A curve that illustrates different bundles of
  goods at different levels of income (ceteris
  paribus) is the Income Offer Curve.
• A plot of quantity demanded against income is
  called an Engel curve.

6
Income Offer and Engel Curves

• How does the value of x1(p1,p2,m) and
  x2(p1,p2,m) change as m changes, holding both p1
  and p2 constant?
• Mapping out the optimal demands for all income
  levels gives the income offer curve.

7
Income Offer and Engel Curves
Fixed p1 and p2.
x2
m lt m lt m
3 budget constraints for 3 different income
levels.
x1
8
Income Offer and Engel Curves
Fixed p1 and p2.
x2
m lt m lt m
x2
x2
x2
x1
x1
x1
x1
9
Income Offer and Engel Curves
Fixed p1 and p2.
x2
m lt m lt m
Incomeoffer curve
x2
x2
x2
x1
x1
x1
x1
10
Income Offer and Engel Curves

• From the income offer curve, we can plot quantity
  demanded on the horizontal axis against income on
  the vertical axis. (Engel curve)

11
Income Offer and Engel Curves
Fixed p1 and p2.
m lt m lt m
x2
Engelcurve good 1
Incomeoffer curve
m
x2
m
x2
m
x2
m
x1
x1
x1
x1
x1
x1
x1
x1
12
Income Offer and Engel Curves

• Now consider special cases
• Cobb-Douglas
• Perfect Complements
• Perfect Substitutes
• Quasilinear

13
Income Offer and Engel Curves Cobb-Douglas

• An example of computing the equations of Income
  Offer curves the Cobb-Douglas case.

14
Income Offer and Engel Curves Cobb-Douglas

• Recall that for Cobb-Douglas preferences, it is
  always the case that at the optimal demands,
  MRSSlope of budget constraint.

15
Income Offer and Engel Curves Cobb-Douglas

• The condition that MRSprice ratio gives an
  equation for the income offer curve.
• Notice that holding prices constant, the income
  offer curve is linear.

16
Income Offer and Engel Curves Cobb-Douglas

• An example of computing the equations of Engel
  curves the Cobb-Douglas case.
• Recall that the ordinary demand equations are

17
Income Offer and Engel Curves Cobb-Douglas

• Solving the demand for income (m), we can get
  equations for the Engel Curves given prices.

Engel Curve for good 1
Engel Curve for good 2
18
Income Offer and Engel Curves Cobb-Douglas
m
Engel curvefor good 1
x1
m
Engel curvefor good 2
x2
For C-D preferences, engel curves are linear.
19
Income Offer and Engel Curves Perfect Complements

• Another example of computing the equations of
  Engel curves the perfectly-complementary case.
• Map out the optimal demands for different income
  levels.

20
Income Offer and Engel Curves Perfect Complements
Fixed p1 and p2.
x2
m lt m lt m
Optimal demands are always along the line
connecting the vertices of the indifference
curves.
x1
21
Income Offer and Engel Curves Perfect Complements
Fixed p1 and p2.
x2
m lt m lt m
Income Offer Curve x1x2
x2
x2
x2
x1
x1
x1
x1
22
Income Offer and Engel Curves Perfect Complements
Fixed p1 and p2.
x2
m lt m lt m
m
x2
m
Engelcurve good 1
x2
m
x2
m
x1
x1
x1
x1
x1
x1
x1
x1
23
Income Offer and Engel Curves Perfect Complements
Engelcurve good 2
m
Fixed p1 and p2.
m
x2
m
m lt m lt m
m
x2
x2
x2
x2
x2
x2
x2
x1
x1
x1
x1
24
Income Offer and Engel Curves Perfect Complements

• One an also compute the equations of Engel
  curves the perfectly-complementary case.
• Recall the ordinary demands take the form

25
Income Offer and Engel Curves Perfect Complements
Rearranged to isolate m
Engel curve for good 1
Engel curve for good 2
26
Income Offer and Engel Curves Perfect Complements
Engelcurve good 2
m
Fixed p1 and p2.
m
m
m
x2
x2
x2
m
x2
m
Engelcurve good 1
m
m
x1
x1
x1
For perfect complements get linear Income Offer
and Engel curves.
x1
27
Income Offer and Engel Curves Perfect Substitutes

• Another example of computing the equations of
  Engel curves the perfectly-substitution case.
• The ordinary demand equations are

28
Income Offer and Engel Curves Perfect Substitutes
29
Income Offer and Engel Curves Perfect Substitutes
Suppose p1 lt p2. Then
and
30
Income Offer and Engel Curves Perfect Substitutes

• Draw the Income Offer curve when p1 lt p2.
• In this case we know that the consumer will only
  consume good 1 no matter what the income level is.

31
Income Offer and Engel Curves Perfect Substitutes
Fixed p1 and p2.
x2
p1 lt p2
Budget Constraints for different income levels.
m lt m lt m
x1
32
Income Offer and Engel Curves Perfect Substitutes
Fixed p1 and p2.
x2
p1 lt p2
Income offer Curve given p1 lt p2
x1
33
Income Offer and Engel Curves Perfect Substitutes
Suppose p1 lt p2. Then
and
and
34
Income Offer and Engel Curves Perfect Substitutes
Given p1 lt p2
m
m
x1
x2
0
Engel curvefor good 1
Engel curvefor good 2
35
Income Offer and Engel Curves Perfect Substitutes

• Notice for perfect substitutes, both income offer
  and Engel curves are linear as well.

36
Homotheticity

• In every example so far the Engel curves and
  income offer curves have all been straight
  lines?Q Is this true in general?
• A No. Engel curves are straight lines if the
  consumers preferences are homothetic.

37
Homotheticity

• If the ratio of the two goods consumed never
  changes with income, the consumer has homothetic
  preferences.
• With homothetic preferences, income offer curves
  and Engel curves will be linear.

38
Homotheticity

• In income goes up, consumption increases
  proportionally.
• The percent change in income the percent change
  in demand.
• Luxury and necessary goods are non-homothetic.

39
Luxury and Necessary goodsnon-homothetic

• A luxury good is one in which demand for a good
  goes up in greater proportion than income.
• ?demand gt ?income
• A necessary good is one in which demand for a
  good goes up by a lesser proportion than income
  went up.
• ?demand lt ?income

40
Income Offer and Engel Curves Quasilinear

• Quasilinear preferences are not homotheticwill
  get non-linear engel curves and income offer
  curves.
• For example,

41
Income Offer and Engel Curves Quasilinear
x2
Each curve is a vertically shifted copy of the
others.
Each curve may intersectboth axes.
x1
42
Income Offer and Engel Curves Quasilinear
For quasilinear preferences (linear in x2), the
optimal demands are such that up to a point,
only x1 is consumed, then as income
increases only x2 consumption is increased.
43
Income Offer and Engel Curves Quasilinear

• At an interior solution,
• MRS-p1/p2

44
Income Offer and Engel Curves Quasilinear

• This implies that as income increases, the
  consumer only increases x1 until
• After this point, only consumption of x2 is
  increased.

45
Income Offer and Engel Curves Quasilinear
x2
Income Offer Curve
x1
46
Income Offer and Engel Curves Quasilinear
x2
Engelcurve forgood 1
m
x1

x1
x1
47
Income Offer and Engel Curves Quasilinear
Engelcurve forgood 2
m
x2
x2
x1
48
Normal and Inferior Goods

• A good for which quantity demanded rises with
  income is called normal.
• Therefore a normal goods Engel curve is
  positively sloped.

49
Normal and Inferior Goods

• A good for which quantity demanded falls as
  income increases is called inferior.
• Therefore an inferior goods Engel curve is
  negatively sloped.

50
Normal and Inferior Goods

• One can also tell if a good is normal or inferior
  from the demand.
• Example Cobb-Douglas Demands increase when
  income increases.

51
Normal and Inferior Goods

• In the following example, good 2 is always normal
  and good 1 becomes inferior.

52
Income Changes Good 2 Is Normal, Good 1 becomes
Inferior
x2
Indifference Curves
Look at optimal choices for different income
levels.
x1
53
Income Changes Good 2 Is Normal, Good 1 becomes
Inferior
x2
x1
54
Income Changes Good 2 Is Normal, Good 1 becomes
Inferior
x2
x1
55
Income Changes Good 2 Is Normal, Good 1 becomes
Inferior
x2
x1
56
Income Changes Good 2 Is Normal, Good 1 becomes
Inferior
x2
x1
57
Income Changes Good 2 Is Normal, Good 1 becomes
Inferior
x2
The demand for good 2 always increases with
income, but the demand for good 1 increases
and then decreases with income.
Incomeoffer curve
x1
58
Income Changes Good 2 Is Normal, Good 1 becomes
Inferior
m
x2
Engel curvefor good 2
x2
m
Engel curvefor good 1
x1
x1
59
Own Price Changes

• Now look at how quantity demanded changes when
  prices change.
• Price offer curve
• Demand curve

60
Price Offer Curve and the Demand Curve

• Want to derive the price offer curve which
  depicts the optimal choices of bundles of goods
  as the price of only one good changes.
• Want to derive the demand for a good as its own
  price changes.

61
Price Offer Curve and the Demand Curve

• How does x1(p1,p2,m) change as p1 changes,
  holding p2 and m constant?
• Suppose only p1 increases, from p1 to p1 and
  then to p1.

62

Price Offer Curve and the Demand Curve
x2
Fixed p2 and m.
Original BC p1x1 p2x2 m
slope-p1/p2
x1
Suppose only p1 increases, from p1 to p1 and
then to p1.
63
Price Offer Curve and the Demand Curve
Fixed p2 and m.
x2
slope-p1/p2
Slope -p1/ p2
x1
64
Price Offer Curve and the Demand Curve
Fixed p2 and m.
x2
slope-p1/p2
Slope -p1/ p2
x1
Slope -p1/ p2
65
Price Offer Curve and the Demand Curve
x2
Fixed p2 and m.
p1 p1
x1
66
Price Offer Curve and the Demand Curve
Fixed p2 and m.
x2
p1 p1
x1
x1(p1)
67
Price Offer Curve and the Demand Curve
p1
Fixed p2 and m.
x2
p1 p1
p1
x1
x1(p1)
x1
x1(p1)
68
Price Offer Curve and the Demand Curve
p1
Fixed p2 and m.
x2
p1 p1
p1
x1
x1(p1)
x1
x1(p1)
x1(p1)
69
Price Offer Curve and the Demand Curve
p1
Fixed p2 and m.
x2
p1
p1
x1
x1(p1)
x1(p1)
x1
x1(p1)
x1(p1)
70
Price Offer Curve and the Demand Curve
p1
Fixed p2 and m.
x2
p1 p1
p1
p1
x1
x1(p1)
x1(p1)
x1
x1(p1)
x1(p1)
x1(p1)
71
p1
Price Offer Curve and the Demand Curve
Fixed p2 and m.
p1
x2
p1
p1
x1
x1(p1)
x1(p1)
x1(p1)
x1
x1(p1)
x1(p1)
x1(p1)
72
p1
Price Offer Curve and the Demand Curve
Ordinarydemand curvefor commodity 1
Fixed p2 and m.
p1
x2
p1
p1
x1
x1(p1)
x1(p1)
x1(p1)
x1
x1(p1)
x1(p1)
x1(p1)
73
p1
Price Offer Curve and the Demand Curve
Ordinarydemand curvefor commodity 1
Fixed p2 and m.
p1
x2
p1
p1 price offer curve
p1
x1
x1(p1)
x1(p1)
x1(p1)
x1
x1(p1)
x1(p1)
x1(p1)
74
Price Offer Curve and the Demand Curve

• The curve containing all the utility-maximizing
  bundles traced out as p1 changes, with p2 and m
  constant, is the p1- price offer curve.
• The plot of the x1 demand against p1 is the
  demand curve for commodity 1.

75
Price Offer Curve and the Demand Curve

• Now look at specific examples
• Cobb-Douglas
• Perfect Complements
• Perfect Substitutes

76
Price Offer Curve and the Demand Curve Cobb
Douglas

• What does a p1 price-offer curve look like for
  Cobb-Douglas preferences?

77
Price Offer Curve and the Demand Curve Cobb
Douglas

• What does a p1 price-offer curve look like for
  Cobb-Douglas preferences?
• TakeThen the ordinary demand functions for
  commodities 1 and 2 are

78
Price Offer Curve and the Demand Curve Cobb
Douglas
and
Notice that x1 gets smaller as p1 increases but
x2 does not vary with p1 so thep1 price offer
curve is flat.
79
Price Offer Curve and the Demand Curve Cobb
Douglas
Fixed p2 and m.
x2
x1
80
p1
Price Offer Curve and the Demand Curve Cobb
Douglas
Ordinarydemand curvefor commodity 1 is
Fixed p2 and m.
x2
x1
x1
81
Price Offer Curve and the Demand Curve Perfect
Complements

• What does a p1 price-offer curve look like for a
  perfect-complements utility function?

82
Price Offer Curve and the Demand Curve Perfect
Complements

• What does a p1 price-offer curve look like for a
  perfect-complements utility function?

Then the ordinary demand functionsfor
commodities 1 and 2 are
83
Price Offer Curve and the Demand Curve Perfect
Complements
84
Price Offer Curve and the Demand Curve Perfect
Complements
With p2 and m fixed, higher p1 causessmaller x1
and x2.
85
Price Offer Curve and the Demand Curve Perfect
Complements
With p2 and m fixed, higher p1 causessmaller x1
and x2.
As
86
Price Offer Curve and the Demand Curve Perfect
Complements
With p2 and m fixed, higher p1 causessmaller x1
and x2.
As
As
87
Price Offer Curve and the Demand Curve Perfect
Complements
Fixed p2 and m.
x2
x1
88
Price Offer Curve and the Demand Curve Perfect
Complements
p1
Fixed p2 and m.
x2
p1 p1
m/p2
p1
x1


x1
89
p1
Price Offer Curve and the Demand Curve Perfect
Complements
Fixed p2 and m.
x2
p1 p1
p1
m/p2
p1
x1

x1
90
p1
Price Offer Curve and the Demand Curve Perfect
Complements
Fixed p2 and m.
p1
x2
p1 p1
p1
m/p2
p1
x1
x1
91
p1
Price Offer Curve and the Demand Curve Perfect
Complements
Ordinarydemand curvefor commodity 1 is
p1
Fixed p2 and m.
x2
p1
m/p2
p1
x1
x1
92
Price Offer Curve and the Demand Curve Perfect
Substitutes

• What does a p1 price-offer curve look like for a
  perfect-substitutes utility function?

Then the ordinary demand functionsfor
commodities 1 and 2 are
93
Price Offer Curve and the Demand Curve Perfect
Substitutes
and
94
Price Offer Curve and the Demand Curve Perfect
Substitutes
Fixed p2 and m.
x2
Know that will only consume good 1.
p1 p1 lt p2
Budget Constraint
x1
95
p1
Price Offer Curve and the Demand Curve Perfect
Substitutes
x2
Fixed p2 and m.
p1 p1 lt p2
p1
x1
x1
96
p1
Price Offer Curve and the Demand Curve Perfect
Substitutes
x2
Fixed p2 and m.
p1 p1 p2
p1
x1
Budget Constraint
x1
97
p1
Price Offer Curve and the Demand Curve Perfect
Substitutes
x2
Fixed p2 and m.
p1 p1 p2
p1
x1
Any point on budget constraint is optimal.
x1
98
p1
Price Offer Curve and the Demand Curve Perfect
Substitutes
Fixed p2 and m.
x2
p1 p1 p2
p2 p1
p1
x1
x1
99
p1
Price Offer Curve and the Demand Curve Perfect
Substitutes
p1
Fixed p2 and m.
x2
p1 p1 gtp2
p2 p1
p1
x1
Only consume Good 2.
x1
100
p1
Ordinarydemand curvefor commodity 1
Price Offer Curve and the Demand Curve Perfect
Substitutes
p1
x2
Fixed p2 and m.
p2 p1
p1 price offer curve
p1
x1
x1
101
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?

102
Own-Price Changes
p1
Given p1, what quantity isdemanded of commodity
1?
p1
x1
103
Own-Price Changes
p1
Given p1, what quantity isdemanded of commodity
1?Answer x1 units.
p1
x1
x1
104
Own-Price Changes

• Taking quantity demanded as given and then asking
  what must be price describes the inverse demand
  function of a commodity.
• One gets the inverse demand by solving for
  price, and one gets the demand by solving for
  quantity demanded.

105
Own-Price Changes
A Cobb-Douglas example
is the ordinary demand function and
is the inverse demand function.
106
Own-Price Changes
A perfect-complements example
is the ordinary demand function and
is the inverse demand function.
107
Ordinary Goods

• A good is called ordinary if the quantity
  demanded of it always increases as its own price
  decreases.

108
Ordinary Goods
Fixed p2 and m.
x2
x1
109
Ordinary Goods
Fixed p2 and m.
As p1 decreases (budget set expands, optimal x1
increases. Good 1 is ordinary.
x2
p1 price offer curve
x1
110
Ordinary Goods
Fixed p2 and m.
Downward-sloping demand curve
x2
p1
p1 price offer curve
Û
Good 1 isordinary
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.

112
Giffen Goods
Fixed p2 and m.
x2
p1 price offer curve
As p1 decreases (budget set expands, optimal x1
at first increases but then decreases. Good 1
becomes Giffen.
x1
113
Ordinary Goods
Demand curve has a positively
sloped part
Fixed p2 and m.
x2
p1
p1 price offer curve
Û
Good 1 becomesGiffen
x1
x1
114
Cross-Price Effects Gross Substitutes and
Complements

• 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.

115
Cross-Price Effects Substitutes and Complements
A perfect-complements example
so
Therefore commodity 2 is a grosscomplement for
commodity 1.
116
Cross-Price Effects Substitutes and Complements
p1
Increase the price ofgood 2 from p2 to p2and
p1
p1
p1
x1
117
Cross-Price Effects Substitutes and Complements
p1
Increase the price ofgood 2 from p2 to
p2causes the demand curve for good 1 to shift
inwards-- good 2 is acomplement for good 1.
p1
p1
p1
x1
118
Cross-Price Effects Substitutes and Complements
A Cobb- Douglas example
so
119
Cross-Price Effects Substitutes and Complements
A Cobb- Douglas example
so
Therefore commodity 1 is neither a
grosscomplement nor a gross substitute
forcommodity 2.