Intermediate Microeconomic Theory
Demand
1
Demand Analysis

In analyzing individual’s behavior
regarding a given good, we start with a
consumer’s demand function for that good.
q1(p1,p2,m)

Recall that a demand function was derived
from “first principles”, or the explicit model
of preferences and choice given a budget
constraint.

However, in the end, it is this derived
demand function that tells us all we need to
know about consumer behavior.
2
Demand Analysis

The first thing we will characterize is how
a consumer’s demand for a given good
changes with his or her income.

Normal Good - demand for good rises with
income, or
q1 ( p1 , p2 , m)
0
m

How would we show this graphically?

Examples of normal goods?
3
Demand Analysis

Alternatively, consider a good where
demand decreases as income increases, or
q1 ( p1 , p2 , m)
0
m

Inferior Good

How would we show this graphically?

Examples of inferior goods?
4
Demand Analysis

Would linear preferences (U(q1,q2) = q1 + q2) imply q1 is a
normal good?

How about quasi-linear preferences (U(q1,q2) = aq10.5 + q2) ?
5
Income Expansion Paths and Engel Curves

We often want to describe how demand changes as income changes.
To do so, we use:



Income Expansion Path - traces out optimal bundle as income
increases, holding prices constant.
Engel Curve – shows how demand for a given good changes as income
changes, holding prices constant.
How do we show these graphically?
6
Deriving Engel Curve Analytically

Straightforward given a utility function and prices.

Example: Suppose prices are p1 = 5, p2 = 10 and
utility function given by U(q1,q2) = q10.5q20.5 .

What will be the equations for his Engel Curve for
good 1 and good 2?

Do these make sense?
7
The Slope of the Engel Curve

The slope of the Engel Curve is
informative.

Consider the Engel Curve shown below.
m
q1

What does linearity imply (sometimes
referred to as homothesticity)?

Do Cobb-Douglas preferences imply such a
linear Engel curve?
8
The Slope of the Engel Curve

The slope of the Engel Curve don’t have to
be constant.

Consider the Engel Curve shown below.
m
q1

What does concave shape imply?

What might be examples of goods with such
an Engel curve? What is a good term to
describe such goods?
9
The Slope of the Engel Curve

Consider the Engel Curve with convex shape.
m
q1

What does convex shape imply (hint: consider
Engel curve for quasi-linear preferences)?

What might be examples of goods with such an
Engel curve? What is a good term to describe
such goods?
10
The Slope of the Engel Curve

What will be the shape of the Engel Curve
for an inferior good?
11
Demand Curves

We also often want to describe how demand changes as prices
change. To do so, we use:



Price Offer Curve - traces out optimal bundle as price of one good
changes, holding other prices and income constant.
Demand Curve – shows how demand for a given good changes as its
price changes, holding other prices and income constant.
How do we show these graphically?
12
Deriving a Demand Curve Analytically

Also straightforward given a utility function,
prices of other goods and income.

Example: Suppose prices are m = 24, p2 = 10 and an
individual has a utility function U(q1,q2) = q10.5q20.5
.

What will be the equation for his Demand Curve for
good 1?

Does this make sense?
13
Demand Curves

Slope of demand curve indicates how
much demand reacts to price.

Generally, demand curves will be downward
sloping, meaning as price rises demand falls,
or
q1 ( p1 , p2 , m)
0
p1

We saw this will be the case with CobbDouglas specification of preferences. How
about with linear utility function?

Given our assumptions, is it possible for a
demand curve to slope upward?
14
Gross Substitutes and Complements

We have already discussed perfect substitutes
and perfect complements.

We can now consider more nuanced definitions
of substitutes and complements, with perfect
versions being subsets.
15
Gross Substitutes and Complements

Beer and pizza.




Generally, I like to consume them together, one
beer with every slice.
However, if the price of a slice went to $10, I
might behave a little differently.
Some degree of complementarity, but not
perfect.
Pizza and Chicken Wings.



Clearly they aren’t perfect substitutes. Even if
pizza was more expensive, I still might order a
few chicken wings.
A raise in the price of one would definitely cause
me to consume less of it an more of the other.
Some degree of substitutability between them
but not perfect.
16
Gross Substitutes and Complements



Gross substitutes - a rise in the
price of one increases the
demand for the other, or if
q1 ( p1 , p2 , m)
0
p2
How would we show this
graphically?
Consider skiing and golf.
Suppose the price of lift tickets
increased.


How will the demand for golf
course time be affected?
How about the demand curve for
golf course time?
17
Gross Substitutes and Complements

Gross complements - a rise in
the price of one decreases the
demand for the other, or if
q1 ( p1 , p2 , m)
0
p2

How would we show this
graphically?

Consider skiing and plane
tickets. Suppose the price of lift
tickets increased.


How will the demand for a plane
ticket to SLC be affected?
How about the demand curve?
18
Gross Substitutes and Complements

If someone’s preferences over two goods
are captured by a Cobb-Douglas Utility
function, will the two goods be gross
substitutes, gross compliments, or neither?
19
Demand Curve and changes in income

Consider the Demand Curve for good 1 for
an individual with Cobb-Douglas Utility
U(q1,q2) = q10.6q20.4 , who has m = 20 and
p2 = 4.


How would this Demand Curve be affected
by a change in endowment from m = 20 to
m = 30?
What if preferences were such that good 1
was an inferior good?
20
Summary

Engel curves and Demand curves are derived from demand
function (which is in turn derived from underlying
preferences).

A Demand curve for a given good describes how the demand
for that good changes as its own price changes, holding other
prices and endowment fixed.


Changes in prices of other goods or endowment may shift a
Demand curve.
An Engel curve for a given good describes how the demand
for that good changes as income changes, holding other prices
fixed.

Changes in prices may shift an Engel curve.
21
Summary


This means that:

The change in demand for good i in reaction to a
change in the price of good i is captured by a
movement along the demand curve for good i.

The change in demand for good i in reaction to
change in price of some other good j or change in
income is captured by a shift in the demand curve
for good i.
Similarly,

The change in demand for good i in reaction to a
change in income is captured by a movement
along the Engel curve for good i.

The change in demand for good i in reaction to
change in price of good i or another good j is
captured by a shift in the Engel curve for good i.
22