Demand and Cross Price Elasticity

1
Demand and Cross Price Elasticity
2
Demand
p1 3 p2 3 I 54
x2
x1
P
Q
3
Demand
p2 3 I 54
x2
x1
P
?
4
?
3
?
2
D
9
Q
3
20
4
Demand
p2 3 I 54
When the price of x1 goes up (from 3 to 4 and an
income of 54), demand for x2 goes up.
x2
x1
P
?
4
?
3
?
2
D
9
Q
3
20
5
Demand
p2 3 I 54
When the price of x1 goes up (from 3 to 4 and an
income of 54), demand for x2 goes up. x1 and x2
are gross substitutes
x2
x1
P
?
4
?
3
?
2
D
9
Q
3
20
6
Definitions

• If the price of one good increases and the demand
  for another decreases, the goods are gross
  complements.
• If the price of one good increases and the demand
  for another increases, the goods are gross
  substitutes.

7
Demand
p2 3 I 54
x2

• Two things happening
• When the price of x1 decreases, your budget set
  is smaller
• Negative income effect
• 2) When the price of x1 decreases, you may ask,
  could I use x2 in place of x1 or do they go
  together.
• Pure substitution effect.

x1
P
?
4
?
3
?
2
D
9
Q
3
20
8
Gross vs. Net

• Gross
• Includes the income effect.
• Net
• Only looks at the pure substitution effect.
• In this class (and most practical applications)
  we care about gross substitution.
• If the price of a good doesnt change much and
  you dont spend too much on it, then the income
  effect is likely small and net gross.

9
Question

• Examples when gross would be a bad approximation
  to net?

10
Calculating Elasticity

• Own-price elasticity

11
Calculating Elasticity

• Own-price elasticity
• Cross-price elasticity

12
Calculating Elasticity

• Cross-price elasticity

13
Demand
p2 3 I 54
x2
Mini-problem could you find the demands for x2
given the income and demand information about x1?
x1
P
?
4
?
3
?
2
D
9
Q
3
20
14
Demand
p2 3 I 54
x2
Mini-problem could you find the demands for x2
given the income and demand information about
x1? A Income expenditure on x1
expenditure on x2
x1
P
?
4
?
3
?
2
D
9
Q
3
20
15
Demand
p2 3 I 54
x2
Mini-problem could you find the demands for x2
given the income and demand information about
x1? A Income expenditure on x1
expenditure on x2 expenditure on x1 p1q1 At
p14, q13, so expenditure equals 12. Income 54
Expenditure on x2 54 12 42 You had 42 to
spend and price was 3 42/3 14.
14
9
4.67
x1
P
?
4
?
3
?
2
D
9
Q
3
20
16
Demand
p2 3 I 54
What is the cross price elasticity of x2 with
respect to x1 when p1 increases from 3 to 4?
x2
14
9
4.67
x1
P
?
4
?
3
?
2
D
9
Q
3
20
17
Demand
p2 3 I 54
What is the cross price elasticity of x2 with
respect to x1 when p1 increases from 3 to 4?
x2
14
9
4.67
x1
P
?
4
?
3
?
2
D
9
Q
3
20
18
Demand
p2 3 I 54
What is the cross price elasticity of x2 with
respect to x1 when p1 increases from 3 to 4?
x2
14
9
4.67
x1
P
?
4
?
3
?
2
D
9
Q
3
20
19

• If two goods are gross substitutes, their
  cross-price elasticity must be positive.
• Why?

20

• If two goods are gross substitutes, their
  cross-price elasticity must be positive.
• Why?

21

• If two goods are gross complements, their
  cross-price elasticity must be negative.
• Why?

22

• If two goods are gross complements, their
  cross-price elasticity must be negative.
• Why?

23
Demand
p2 3 I 54
x2
14
9
4.67
x1
P2
P1
?
4
?
?
?
?
3
3
?
2
D1
9
4.67
14
Q2
9
Q1
3
20
24
Demand
p2 3 I 54
A change in the price of one good can change
both the quantity demanded demand
x2
14
9
4.67
x1
P1 4
P2
P1
P1 3
P1 2
?
4
?
?
?
?
3
3
?
D2
2
D1
D2
D2
9
4.67
14
Q2
9
Q1
3
20
25
Review

• Cross price demand elasticity is one way to
  measure the responsiveness to changes in the
  price of other goods.
• Goods are gross complements if an increase in the
  price of one lowers demand for the other.
• Goods are gross substitutes if an increase in the
  price of one increases demand for the other.
• An increase in price involves both an income
  effect and a pure substitution effect.
• If there are many goods and the good with the
  price change does not take up too much of
  expenditure, net gross.
• Gross substitutes have positive cross price
  elasticities
• Gross complements have negative cross price
  elasticities.

26
Demand and Income Elasticty
27
Preferences
x2
x1
28
Prices and Income
p1 3 p2 3 I 54
x2
18
?
x1
18
29
Income changes
p1 3 p2 3 I 60
x2
20
18
?
?
x1
18
20
30
Income changes again
p1 3 p2 3 I 52
x2
20
18
17
?
?
?
x1
18
20
17
31
Engel Curve
p1 3 p2 3 I 52
x2
?
?
?
x1
As your income increases, demand for both goods
increases. Both x1 and x2 are normal goods.
32
Christian Lorenz Ernst Engel, 1821-1896
33
Preferences
x2
x1
34
Income and prices
p1 3 p2 3 I 48
x2
16
?
x1
16
35
Income changes
p1 3 p2 3 I 60
x2
20
16
?
?
x1
16
20
36
Income changes again
p1 3 p2 3 I 30
x2
20
16
10
?
?
?
x1
16
20
10
37
Engel Curve
p1 3 p2 3
x2
?
?
?
x1
38
Engel Curve
p1 3 p2 3
x2
?
?
?
x1
As your income increases, demand for x1
increases.
39
Engel Curve
p1 3 p2 3
x2
?
?
?
x1
As your income increases, demand for x1
increases. As your income increases, demand for
x2 increases, then decreases
40
Engel Curve
p1 3 p2 3
x2
?
?
?
x1
As your income increases, demand for x1
increases. As your income increases, demand for
x2 increases, then decreases Both x1 and x2 are
normal goods at low income, but x2 is
inferior at high income.
41
Engel Curves
x2
x1
Linear Engel Curve If income doubles, demand for
x1 doubles. Since p is constant, expenditure on
x1 doubles. Expenditure on x1 is a constant
proportion of income.
42
Engel Curves
x2
x1
Arching down If income doubles, demand for x1
more than doubles. Since p is constant,
expenditure on x1 more than doubles. Expenditure
on x1 is an increasing proportion of income. ?
luxury
43
Engel Curves
x2
x1
Arching up If income doubles, demand for x1 less
than doubles. Since p is constant, expenditure on
x1 less than doubles. Expenditure on x1 is a
decreasing proportion of income. ? necessity
44
Engel Curves
x2
x1
Bending back If income increases, demand for x1
decreases. Expenditure on x1 is decreasing in
income ? inferior.
45
Importance to Agriculture

• Engels Law

"... je aermer eine Familie ist, einen desto
groesseren Antheil von der Gesamtausgabe muss zur
Beschaffung der Nahrung aufgewendet werden ..."
46
Importance to Agriculture

• Engels Law

As income increases, households spend a smaller
percentage of income on food.
47
Income elasticity
p1 3 p2 3
x2
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
48
Compare with price elasticity
49
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x1 when
increases from 30 to 48
50
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x1 when
increases from 30 to 48
51
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x1 when
increases from 30 to 48
52
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x1 when
increases from 30 to 48
When income goes up by 1, demand for x1 goes up
by 1.18. At an income of 30, x1 is a luxury
good.
53
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x2 when
increases from 30 to 48
54
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x2 when
increases from 30 to 48
55
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x2 when
increases from 30 to 48
56
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x2 when
increases from 30 to 48
When income goes up by 1, demand for x2 goes up
by.84. At an income of 30, x2 is a necessity.
57
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x2 when
increases from 48 to 60
58
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x2 when
increases from 48 to 60
59
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x2 when
increases from 48 to 60
60
Income elasticity
x2
p1 3 p2 3
I60
I48
I30
?
9
?
7
?
6
x1
7
13
4
Example What is the income elasticity of x2 when
increases from 48 to 60
When income goes up by 1, demand for x2 goes
down by 1.13. At an income of 30, x2 is an
inferior good.
61
Comparison
Engel Curve
Income Elasticity
Linear Engel Curve
If income goes up by z, demand for x goes up by
z
62
Comparison
Engel Curve
Income Elasticity
x2
x1
Downward arching Engel Curve
If income goes up by z, demand for x goes up by
more than z. Luxury good
63
Comparison
Engel Curve
Income Elasticity
x2
x1
Upward arching Engel Curve
If income goes up by z, demand for x goes up by
less than z. Necessity good
64
Comparison
Engel Curve
Income Elasticity
x2
x1
Backward bending Engel Curve
If income goes up by z, demand for x goes
down. Inferior good
65
Warning

• Own price elasticities often neglect the negative
  sign because it is (almost) always negative, i.e.
  no information gained by including it
• The negative sign is important information about
  the income elasticity.

66
Importance to Agriculture

• Engels Law

As income increases, households spend a smaller
percentage of income on food.
67
Importance to Agriculture

• Engels Law

The income elasticity of food is positive but
less than 1.
68
Back to Demand
x2
P2
p1 3 p2 3
I60
p1 3
I48
I30
I30
?
9
?
3
?
7
?
6
D
x1
7
13
4
Q2
6
69
Back to Demand
x2
P2
p1 3 p2 3
I60
p1 3
I48
I30
I30
?
9
?
3
?
7
?
6
D
x1
7
13
4
Q2
6
Suppose income increased to 48?
70
Back to Demand
x2
P2
p1 3 p2 3
I60
p1 3
I48
I30
I30
?
9
?
3
?
?
7
?
6
D
x1
7
13
4
Q2
6
9
Suppose income increased to 48?
71
Back to Demand
x2
P2
p1 3 p2 3
I60
p1 3
I48
I30
I30
?
9
?
3
?
?
7
?
6
I48
D
D
x1
7
13
4
Q2
6
9
Suppose income increased to 48? Must be on a new
demand curve! There was a change in both the
quantity demanded and demand
72
Back to Demand
x2
P2
p1 3 p2 3
I60
p1 3
I48
I30
I30
?
9
?
3
?
?
7
?
6
I48
D
D
x1
7
13
4
Q2
6
9
Suppose income increased to 48? Must be on a new
demand curve!
Calculated before For x2, from I30 to I48
73
Back to Demand
x2
P2
p1 3 p2 3
I60
p1 3
I48
I30
I30
?
9
?
3
?
?
7
?
6
I48
D
D
x1
7
13
4
Q2
6
9
Suppose income increased to 48? The good is
normal between 30 and 48, and thus demand shifted
out(up).
74
Back to Demand
x2
P2
p1 3 p2 3
I60
p1 3
I48
I30
I30
?
9
?
3
?
?
?
7
?
6
I48
D
D
x1
7
13
4
Q2
6
9
7
Suppose income increased again to 60?
Calculated before For x2, from I48 to I60
75
Back to Demand
x2
P2
p1 3 p2 3
I60
p1 3
I48
I60
I30
I30
?
9
?
3
?
?
?
7
?
6
I48
D
D
x1
D
7
13
4
Q2
6
9
7
Suppose income increased again to 60? The good
is inferior between 48 and 60, and thus demand
shifted in(down).
76
Review

• Income elasticity is a way to quantify
  responsiveness to changes in income
• If income elasticity is positive, the good is
  normal and demand and expenditure increase in
  income.
• If income elasticity is negative, the good is
  inferior and demand decreases in income.
• If a good is a luxury, its expenditure share
  increases in income
• If a good is a necessity, its expenditure share
  decreases in income (but expenditure still
  increases).
• A change in income can change the quantity
  demanded at any price and thus change demand.
• Engels law is equivalent to saying food is a
  necessity.