Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Econ 401 Price Theory
Chapter 6: Demand
Instructor: Hiroki Watanabe
Summer 2009
1 / 57
Intro
Overview
1
2
3
4
5
6
7
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Introduction
Overview
Comparative Statics
Endogenous & Exogenous Variables
Overview
Marshalian Demand Function
Review
The Nature of Commodities & Graphical
Representation
Income Changes
Normal & Income-Inferior Goods
Income Expansion Path
Example 1: Cobb-Douglas
Example 2: Quasi-Linear Preferences
Example 3: Perfect Substitutes
Example 4: Perfect Compliments
Own-Price Changes
Law of Demand & Giffen Good
Price Offer Curve & Demand Curve
Example 1: Perfect Compliments
Example 2: Perfect Substitutes
Example 3: Cobb-Douglas
Cross-Price Changes
Inverse Demand
Summary
2 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Overview
Chpt 6: Categories and definitions.
Chpt 8: Price change → demand.
Chpt 14: Price change → welfare.
3 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Comparative Statics
Discussion: Situational Change & Consumer’s Reaction
marketplace.publicradio.org/display/web/
2008/06/09/charities/
With the tools we have learned so far, propose a
way to measure or predict "a big dent in
volunteerism" triggered by the increasing gas price.
4 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Comparative Statics
It is necessary (and critical) to observe the change
in demand x∗ (endogenous) caused by given
parameters p and m (exogenous).
Comparative Statics
Comparative statics analysis compares the values of
endogenous variables at the differenct values of the
exogenous variable while other parameters held
constant.
ceteris paribus.
Why can’t we move multiple parameters at the
same time like in the real world?
Prediction becomes less clear-cut (and confusing):
∗ : 3 & 2.
pC : 2 % 5 ceteris paribus ⇒ xC
∗ : 3 → indefinite.
pC : 2 % 5 while pT : 1 % 3 ⇒ xC
1
2
5 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Endogenous & Exogenous Variables
Endogenous & exogenous variables:
Given , p and m, UMP predicts which x∗ Greg will
choose.
UMP does not predict the movement of p, m and
change in tastes .
()
p
UMP
∗
m
6 / 57
Σ
Intro
Overview
1
2
3
4
5
6
7
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Introduction
Overview
Comparative Statics
Endogenous & Exogenous Variables
Overview
Marshalian Demand Function
Review
The Nature of Commodities & Graphical
Representation
Income Changes
Normal & Income-Inferior Goods
Income Expansion Path
Example 1: Cobb-Douglas
Example 2: Quasi-Linear Preferences
Example 3: Perfect Substitutes
Example 4: Perfect Compliments
Own-Price Changes
Law of Demand & Giffen Good
Price Offer Curve & Demand Curve
Example 1: Perfect Compliments
Example 2: Perfect Substitutes
Example 3: Cobb-Douglas
Cross-Price Changes
Inverse Demand
Summary
7 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Marshalian Demand Function
Marshalian Demand Function
Marshalian demand function returns the optimal bundle
at each given price and income, denoted by
€
Š
ϕ(p1 , p2 , m) = ϕ1 (p1 , p2 , m), ϕ2 (p1 , p2 , m) .
()
∗
= ϕ1 (p, m)
1
p
ϕ(p, m)
∗
= ϕ2 (p, m)
2
m
8 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Review
Change in Parameters
Change in price causes rotation to the budget
constraint.
Change in income causes parallel shifts to the
budget constraint.
9 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
The Nature of Commodities & Graphical Representation
Lots of definitions to learn today to describe and
classify different types of commodities.
10 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
The Nature of Commodities & Graphical Representation
↑ ϕ1 (p, m)
↓ ϕ1 (p, m)
↑m
normal income-inferior
↓ p1
LOD
Giffen
↑ p2 substitute
complement
11 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
The Nature of Commodities & Graphical Representation
on x1 − x2 on x1 − parameter
m income expansion path
Engel curve
p1
price offer curve
demand curve
12 / 57
Σ
Intro
Overview
1
2
3
4
5
6
7
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Introduction
Overview
Comparative Statics
Endogenous & Exogenous Variables
Overview
Marshalian Demand Function
Review
The Nature of Commodities & Graphical
Representation
Income Changes
Normal & Income-Inferior Goods
Income Expansion Path
Example 1: Cobb-Douglas
Example 2: Quasi-Linear Preferences
Example 3: Perfect Substitutes
Example 4: Perfect Compliments
Own-Price Changes
Law of Demand & Giffen Good
Price Offer Curve & Demand Curve
Example 1: Perfect Compliments
Example 2: Perfect Substitutes
Example 3: Cobb-Douglas
Cross-Price Changes
Inverse Demand
Summary
13 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Normal & Income-Inferior Goods
Normal & income-inferior Goods
A commodity is called a normal good if increase in
income leads to increase in consumption while other
things being equal. Otherwise, a commodity is
income-inferior.
Does an income-inferior good really exist?
14 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Normal & Income-Inferior Goods
15 / 57
Intro
Overview
Income Changes
Normal & Income-Inferior Goods
Own-Price Changes
Cross-Price Changes
Inverse Demand
Figure:
Mass-produced cheesecakes (2 days past the
expiration date) & fresh home-made cheesecakes
16 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Income Expansion Path
Income Expansion Path
Income expansion path represents the collection of
optimal bundles on x1 − x2 plane under varying income
levels while p = (p1 , p2 ) is held constant.
Engel Curve
A plot of quantity demanded agains income, with fixed
p = (p1 , p2 ) is called an Engel curve
17 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Income Expansion Path
18 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 1: Cobb-Douglas
Cobb-Douglas Utility Function
maxxC ,xT u(xC , xT ) = xC xT2
MRT at (xC , xT ) is
s.t. xC + xT = m.
−xC
.
2xT
19 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Example 1: Cobb-Douglas
€
Š €
Š
2m
ϕ(m, p = (1, 1)) = ϕ1 (m, p), ϕ2 (m, p) = m
,
.
3
3
Engel curve for cheesecakes at p = (1, 1): m = 3xC∗ .
20 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 1: Cobb-Douglas
Income Expansion Path
10
indifference curves
m=3
m=6
m=9
9
8
7
xT
6
5
4
3
2
1
0
0
1
2
3
4
5
xC
6
7
8
9
10
21 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Example 1: Cobb-Douglas
Engel Curve
30
27
24
21
m
18
15
12
9
6
3
0
0
1
2
3
4
5
xC
6
7
8
9
10
22 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 2: Quasi-Linear Preferences
Consider a bundle of coins xC and tea xT .
Quasi-Linear Preferences
p
maxxC ,xT u(xC , xT ) = xC + xT s.t. xC + .5xT = m.
p
MRS at (xC , xT ) is −2 xT .
ϕ(m, p = (1, .5)) = (m − 1, 2).
Recall MRS is same at each xT regardless of the
amount of xC .
23 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Example 2: Quasi-Linear Preferences
Income Expansion Path
10
indifference curves
m=2
m=4
m=6
9
8
7
xT
6
5
4
3
2
1
0
0
1
2
3
xC
4
5
6
24 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 2: Quasi-Linear Preferences
Engel Curve for Coins
7
6
5
m
4
3
2
1
0
0
1
2
3
xC
4
5
6
25 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Example 2: Quasi-Linear Preferences
Engel Curve for Tea
6
5
m
4
3
2
1
0
0
1
2
3
xT
4
5
6
26 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 3: Perfect Substitutes
Consider a bundle of six-packs and bottles of
Corona (x6 , x1 ).
Suppose p = (p6 , p1 ) = (10, 2).
MRS is −6 while the relative price is −5.
27 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Example 3: Perfect Substitutes
Income Expansion Path
48
40
x1
52
44
36
32
28
24
16
20
12
12
indifference curves
m=20
m=40
m=60
48
40
20
18
56
44
36
32
28
24
24
36
32
28
24
16
6
40
20
12
4
5
36
3
x6
32
2
28
1
24
8
16
4
0
0
6
28 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 3: Perfect Substitutes
Engel Curve for Six−Packs
30
m
20
10
0
0
1
2
3
x6
29 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Example 3: Perfect Substitutes
Engel Curve for Bottles
6
5
m
4
3
2
1
0
0
1
2
3
x1
4
5
6
30 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 4: Perfect Compliments
Discussion
Consider a bundle (cereal, milk)= (xC , xM ).
Greg says he can’t have cereals without milk and
the only time he has milk is when he eats his
cereals.
Greg’s preferred cereal-milk ratio is 1 to 1.
(pC , pM ) = (2, 2).
What do Greg’s Engel curves look like? (curve?
flat?)
Consider m = 4 and m = 8 for example.
31 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Example 4: Perfect Compliments
Income Expansion Path
6
5
3
2
1
indifference curves
m=4
m=8
5
4
2
2
1
0
0
x
3
3
2
3
4
1
M
4
1
1
2
1
2
3
xC
1
4
5
6
32 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 4: Perfect Compliments
Engel Curve for Cereal
24
20
m
16
12
8
4
0
0
1
2
3
xC
4
5
6
33 / 57
Intro
Overview
1
2
3
4
5
6
7
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Introduction
Overview
Comparative Statics
Endogenous & Exogenous Variables
Overview
Marshalian Demand Function
Review
The Nature of Commodities & Graphical
Representation
Income Changes
Normal & Income-Inferior Goods
Income Expansion Path
Example 1: Cobb-Douglas
Example 2: Quasi-Linear Preferences
Example 3: Perfect Substitutes
Example 4: Perfect Compliments
Own-Price Changes
Law of Demand & Giffen Good
Price Offer Curve & Demand Curve
Example 1: Perfect Compliments
Example 2: Perfect Substitutes
Example 3: Cobb-Douglas
Cross-Price Changes
Inverse Demand
Summary
34 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Law of Demand & Giffen Good
Law of Demand & Giffen Good
A commodity is said to satisfy the law of demand if the
price increase leads to reduction in quantity demanded.
Otherwise, the commodity is called a Giffen good.
Examples of Giffen goods?
Prada shoes: If they’re cheap and everyone’s
wearing them, then ...
35 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Law of Demand & Giffen Good
36 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Law of Demand & Giffen Good
37 / 57
Intro
Overview
Income Changes
Own-Price Changes
Figure:
Cross-Price Changes
Inverse Demand
Price Offer Curve & Demand Curve
Price Offer 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.
Marshalian Demand Curve
The plot of the x1 -coordinate of the p1 -price offer curve
against p1 is the Marshalian demand curve for
commodity 1.
38 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Price Offer Curve & Demand Curve
39 / 57
Intro
Overview
Income Changes
Own-Price Changes
Figure:
Cross-Price Changes
Inverse Demand
Price Offer Curve & Demand Curve
40 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 1: Perfect Compliments
Perfect complements (x1 , x2 ) with (p1 , p2 ) = (1, 1)
41 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Example 1: Perfect Compliments
42 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 2: Perfect Substitutes
Consider Coke and Pepsi.
43 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Example 2: Perfect Substitutes
44 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 3: Cobb-Douglas
Now consider cheesecakes and tea (xC , xT ):
Cobb-Douglas Utility Function
maxxC ,xT u(xC , xT ) = xC xT
MRS at (xC , xT ) is
s.t. xC + 2xT = m.
−xC
.
x1
45 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Example 3: Cobb-Douglas
pC−Price Offer Curve
75
50
10
C
5
0
10
8
0
p =2
C
25
7
12
5
75
50
6
xM
17
indifference curves
5
p =1
15
12
9
10
0
5
75
4
50
25
3
50
2
25
25
1
0
0
2
4
6
8
10
xC
12
14
16
18
20
46 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Example 3: Cobb-Douglas
Marshalian Demand Curve for Cheesecakes
10
C
9
φ (m=20, pC, pT=2)=20/pC
8
φC (m=10, p , p =2)=10/p
C
T
C
7
pC
6
5
4
3
2
1
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
xC
47 / 57
Intro
Overview
1
2
3
4
5
6
7
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Introduction
Overview
Comparative Statics
Endogenous & Exogenous Variables
Overview
Marshalian Demand Function
Review
The Nature of Commodities & Graphical
Representation
Income Changes
Normal & Income-Inferior Goods
Income Expansion Path
Example 1: Cobb-Douglas
Example 2: Quasi-Linear Preferences
Example 3: Perfect Substitutes
Example 4: Perfect Compliments
Own-Price Changes
Law of Demand & Giffen Good
Price Offer Curve & Demand Curve
Example 1: Perfect Compliments
Example 2: Perfect Substitutes
Example 3: Cobb-Douglas
Cross-Price Changes
Inverse Demand
Summary
48 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
The initial question: how does the increase in the
gas price affect Greg’s spending on community
services (≈ "purchase" of volunteering work).
Substitutes & Compliments
1
A cheesecake is a complement to tea if increase in the
price of tea leads to decrease in cheesecake
consumption.
2
A cheesecake is a substitute to tea if increase in the
price of tea leads to increase in cheesecake
consumption.
Bottles & six-packs of Corona?
Cereal & milk?
Cobb-Douglas utility?
49 / 57
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
(cont’d from the previous example)
max u(xC , xT ) = xC xT
s.t.
xC ,xT
xC + 2xT = m.
ϕT (m = 20, pC , pT = 2) = 10 is independent of the
price of cheesecakes.
Marshalian Demand Curve for Cheesecakes
10
C
9
φ (m=20, pC, pT=2)=20/pC
8
φC (m=10, p , p =2)=10/p
C
T
C
7
6
pC
Intro
5
4
3
2
1
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
xC
50 / 57
Σ
Intro
Overview
1
2
3
4
5
6
7
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Introduction
Overview
Comparative Statics
Endogenous & Exogenous Variables
Overview
Marshalian Demand Function
Review
The Nature of Commodities & Graphical
Representation
Income Changes
Normal & Income-Inferior Goods
Income Expansion Path
Example 1: Cobb-Douglas
Example 2: Quasi-Linear Preferences
Example 3: Perfect Substitutes
Example 4: Perfect Compliments
Own-Price Changes
Law of Demand & Giffen Good
Price Offer Curve & Demand Curve
Example 1: Perfect Compliments
Example 2: Perfect Substitutes
Example 3: Cobb-Douglas
Cross-Price Changes
Inverse Demand
Summary
51 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Inverse Demand Function
Inverse demand function D(x1 ) assigns the price at
which the amount x1 will be chosen, given p2 and m.
52 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
∗
C
(pC , pT )
ϕ(pC , pT , m)
∗
T
m
pC
C
D(C )
m̄, p¯T
53 / 57
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
So?
Normalize p2 = 1.
Tangency condition:
p1 = MRS(x1∗ , x2∗ ).
⇒ D(x1∗ ) = p1 = MRS(x1∗ , x2∗ ) (=marginal willingness
to pay)
Inverse demand function denotes the marginal
willingness to pay at each x1 given p2 and m.
Note the inverse demand function shifts left and
right if you change p2 and m.
54 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Σ
Inverse Demand for Cheesecakes
10
D(xC)=20/xC at m=20
9
D(x )=10/x at m=10
C
8
C
7
pC
6
5
4
3
2
1
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
xC
55 / 57
Intro
Overview
1
2
3
4
5
6
7
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Introduction
Overview
Comparative Statics
Endogenous & Exogenous Variables
Overview
Marshalian Demand Function
Review
The Nature of Commodities & Graphical
Representation
Income Changes
Normal & Income-Inferior Goods
Income Expansion Path
Example 1: Cobb-Douglas
Example 2: Quasi-Linear Preferences
Example 3: Perfect Substitutes
Example 4: Perfect Compliments
Own-Price Changes
Law of Demand & Giffen Good
Price Offer Curve & Demand Curve
Example 1: Perfect Compliments
Example 2: Perfect Substitutes
Example 3: Cobb-Douglas
Cross-Price Changes
Inverse Demand
Summary
56 / 57
Σ
Intro
Overview
Income Changes
Own-Price Changes
Cross-Price Changes
Inverse Demand
Characterize commodities by changing situational
background.
Income changes and associated graphs.
Own-price changes and associated graphs.
Cross-price changes and associated graphs.
Meaning of the inverse demand function.
57 / 57
Σ