Demand
• Using the tools we have developed up to this
point, we can now determine demand for an
individual consumer
Demand and Comparative Statics
• We seek demand functions
– x1* = x1( p1, p2, m) and
– x2* = x2( p1, p2, m)
ECON 370: Microeconomic Theory
• From our previous example
– Preferences of the form: u = xαy1 – α
– And a budget constraint: pxx + pyy ≤ m
– Result in demands:
Summer 2004 – Rice University
Stanley Gilbert
• x* = αm / px
• y* = (1 – α)m / py
Econ 370 - Ordinal Utility
2
Marshallian Demand
Graphically
• In general, we are interested in tracing out
Marshallian Demand Curves.
Preferences
x2
• A Marshallian Demand Curve describes how
demand for a good changes:
Demand for Good 1
– As its own price changes, and
– Holding all other prices and income constant
• Functionally, that means graphing
– x1* = x1( p1, p2, m)
– Versus p1
– And holding p2 and m constant
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
p13
p12
p11
x1
P3
p1
p12
p11
Q
4
1
Aggregating Demand
Aggregating Demand (cont)
• Consumer i’s ordinary demand function for good j is
• Ultimately, we want to look at market demand
– xji( p1, p2, mi)
– (notice that we make allowance for different consumers
having different income)
• That is, we want to see how the demands for many
people add together to make up market demand
• If we have n people
• Then aggregate demand for good xj is:
– And let i = 1 … n
• Two goods, x1, and x2
– Xj( p1, p2, m1, …, mn) = Σi=1n xji( p1, p2, mi)
…
• If all consumers are identical, then
– Xj( p1, p2, M) = n × xj( p1, p2, m)
– Where M = n × m
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
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Aggregating Demand Graphically
Aggregating Demand Graphically
• Graphically, we add demand curves holding price
constant
• Graphically, we add demand curves holding price
constant
– That is, horizontally
Consumer 1
P
Consumer 2
P
Q1
Econ 370 - Ordinal Utility
– That is, horizontally
Market
Consumer 1
P
Q2
P
Qm
7
Consumer 2
P
Q1
Econ 370 - Ordinal Utility
Market
P
Q2
Qm
8
2
The “Law” of Demand
The “Law” of Demand (cont)
• What, in Economics is called the “Law” of
Demand says
• The “Law” of Demand is not a law in the sense
that it is required by Economic theory
– Demand curves “always” slope downward
– That is, Demand for a good “always” decreases as the
price for that good increases (holding everything else
constant)
• A good that does not obey the law of demand is
theoretically possible and is called a Giffen Good
– (A good that obeys the law of demand is called an
Ordinary Good)
• This statement is different from what is in your
book—which I will discuss in due time—but it is
the more usual statement of the law of demand
• However, so far, no one has ever identified a
Giffen good
• So the “Law” of demand is a good description of
empirical reality
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
Elasticity of Demand
10
Calculating Elasticity
• One question we are interested in is how responsive
demand is to changes is price
• There are two types of Elasticity:
• We use something called Elasticity to measure it
– Point Elasticity:
ε=
• In general, Elasticity is the ratio of the percent change of
(here) demand to percent change in price
dxi xi pi dxi
=
dpi pi xi dpi
– Arc Elasticity:
ε=
pi ∆xi pi1 + pi2 xi2 − xi1
=
xi ∆pi xi1 + xi2 pi2 − pi1
• That is:
• We will almost always use Point Elasticity
% ∆xi
εi =
% ∆pi
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
12
3
Elasticity in General
Observations on Demand Elasticity
• Own-Price Elasticity may be different at different points
on the demand curve
• There are many different Elasticities
• Own-Price Elasticity measures how demand
changes as its own price changes
• Since demand generally decreases as price increases, we
have ε < 0
– But usually for convenience we work with | ε |.
• Income Elasticity measures how demand changes
as income changes
• We call demand Elastic if | ε | > 1
• We call demand Inelastic if | ε | < 1
• Cross-Price Elasticities measure how demand
changes as the prices of other goods change
• In general,
– when demand is elastic, it responds strongly to changes in price
– when demand is inelastic, it responds weakly to changes in price
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
Example: Linear Demand Curve
14
Example: Points of Interest
• Suppose inverse demand is linear
pi = a – bxi
– pi = a – bxi
– xi = a/b – pi / b
pi
• We calculate dxi /dpi= -1 / b
a
Observe that
ε = -∞
• Own-price elasticity of demand is
ε=
pi dxi
pi
⎛− 1⎞
=
(a − pi ) b ⎜⎝ b ⎟⎠
xi dpi
=−
pi
a − pi
a/2
ε = -1
pi
a − pi
0
=0
a −0
a2
= −1
⇒ ε =−
a−a 2
a
= −∞
⇒ ε =−
a−a
⇒ ε =−
pi = 0
pi =
ε =−
a
2
pi = a
ε=0
a / 2b
Econ 370 - Ordinal Utility
a/b
xi
15
4
Example: Summary
Important Special Cases
• What is the elasticity of a horizontal demand curve?
pi
a ε
= −∞
own-price elastic
ε = −1
a/2
own-price unit elastic
• What is the elasticity of a vertical demand curve
own-price inelastic
ε =0
a/2b
a/b
xi
Econ 370 - Ordinal Utility
Income Elasticity of Demand
Income Effects Graphically
• We are also interested in demand changes as
income changes
Preferences
Engle Curve
x2
• We use Income Elasticity to measure it, where
m
m3
– Income Elasticity for Good i is defined as:
ε im
18
m2
m dxi
=
xi dm
m3
m2
m1
m1
x1
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
x1
20
5
Graphical Income Elasticity (cont)
Preferences
Normal and Inferior Goods
• For most goods demand increases as income
increases
Engle Curve
x2
m
Income Offer
Curve
– That is εim > 0
– And the Engle curve is sloped forward
– Such goods are called Normal Goods
m3
m2
• However, demand for some goods decreases as
income increases
m1
m1
m2
m3
x1
Econ 370 - Ordinal Utility
x1
21
Inferior to What?
• Note that some goods may be both Normal and
Inferior in different price ranges
Econ 370 - Ordinal Utility
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Normal, Inferior and Giffen Goods
• What exactly is an inferior good, and what is it
inferior to?
• A Normal Good cannot be a Giffen Good
• -or- All Giffen Goods are Inferior goods
• Example: the Yugo
–
–
–
–
–
–
– That is εim < 0
– And the Engle curve is sloped backward
– Such goods are called Inferior Goods
• (Can you see why?)
Very cheap car: Sold for ~ $4,000 in the US in 1986
Poor quality and very small
Sold moderately well in the US
Still in production in Serbia
Who do you think bought the cars?
What do you think happened when their income
increased?
• However, most Inferior Goods are not Giffen
Goods
• Inferior to What?
– To other goods available
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
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6
Can all goods be inferior?
Cross-Price Elasticity of Demand
• Finally, we are interested in how demand changes
as other prices change
• All goods cannot be Inferior, that is
• At least one good must be Normal
• Why?
• We use Cross-Price Elasticity to measure it, where
x2
– Cross-Price Elasticity for Good i relative to the price of
good j is defined as:
ε ij =
p j dxi
xi dp j
m2
m1
x1
• Are there any preferences that will result in consumption
decreasing for both goods when income increases?
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
Substitutes and Complements
26
Examples
• If consumers see two goods as close substitutes for
each other, what sign will the cross-price elasticity
of demand have?
• Perfect Complements/Substitutes
• Homothetic Preferences
• x = Apε
– εij > 0
– What makes this interesting?
• If consumers see two goods as complements for
each other, what sign will the cross-price elasticity
of demand have?
– εij < 0
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
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