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APPLIED ECONOMICS
 REVIEW
 GO OVER HOMEWORK S ET #3
 CONTINUE CONSUMER BEHAVIOR
Derivation of the consumer demand function
 As in the previous example (i.e., changing
the price of one commodity and finding the new consumer
equilibrium point), if we continue to change , we can get
the following:
We see that as
p1 changes,
ceteris paribus,
 shift in the
budget line
 new
consumer
equilibrium
point
Consumer Demand Function
Using these consumer equilibrium points,
we can derive the consumer’s demand for z1.
Demand
illustrates the
quantities of a
good
(commodity)
consumer would
be willing to
purchase at
alternative
prices, ceteris
paribus.
Consumer Demand Function
Mathematical Derivation of Demand
• A consumer’s ordinary demand function
(called the Marshallian demand function)
is derived from utility maximization subject
to a budget constraint.
Mathematical Derivation of Demand
 constrained objective function
Mathematical Derivation of Demand
Mathematical Derivation of Demand
Likewise for z1 , we obtain:
Note that these demand functions are a special case
since they’re functions of only own price and income.
Question:
Are these demand functions downward sloping?
How can you tell?
Downward sloping  the slope is negative
Mathematical Derivation of Demand
The demand for
(Using
reciprocals or
inverses)
Mathematical Derivation of Demand
We can easily solve for
from the demand function:
namely
function is negative or demand is downward sloping.
Mathematical Derivation of Demand
Likewise we get the same result for the consumer demand
for z1… since
negative or demand is downward sloping.
So we have the following graph:
The law of demand
states that price
and quantity that
are demanded are
negatively related.
Mathematical Derivation of Demand
Is income a positive shifter of demand?
Mathematical Derivation of Demand
In the usual case for demand derived from utility
maximization, we have:
Mathematical Derivation of Demand
To derive this general form, we need to adjust the form
of the utility function.
Suppose we have:
Mathematical Derivation of Demand
Mathematical Derivation of Demand
Mathematical Derivation of Demand
Mathematical Derivation of Demand
Mathematical Derivation of Demand
we can rewrite the equation as:
Mathematical Derivation of Demand
Demand function is downward sloping:
Is income a positive shifter of the demand function?
Demand vs. Quantity Demanded
Distinction between demand and quantity demanded:
[This distinction is usually heavily emphasized in
introductory and intermediate microeconomics courses.]
Demand refers to the entire schedule.
Quantity demanded refers to the quantity purchased by the
consumer at a particular price level.
Demand Function
General form of the demand function:
where
our usual demand curve.
If these other factors change, then demand will shift.
Factors Affecting Demand:
1. Changes in income (for normal goods)
(vs inferior or Giffen goods)
Factors Affecting Demand:
1. Changes in income (for normal goods)
2. Changes in the price of substitutes
Take the case of beef and pork (substitutes)
If price of pork ↑  demand for beef ↑
Why? Consumers substitute beef for pork when
the price of pork ↑
If the price of pork ↓ demand for beef ↓
Factors Affecting Demand:
1. Changes in income (for normal goods)
2. Changes in the price of substitutes
3. Changes in the price of complements.
Take the case of milk and cereal (complements):
If the price of milk ↑  demand for cereal ↓.
Why? Milk is an input into cereal consumption.
Likewise, if the price of milk ↓  demand for cereal ↑.
Factors Affecting Demand:
1. Changes in income (for normal goods)
2. Changes in the price of substitutes
3. Changes in the price of complements.
4. Changes in tastes and preferences.
Factors Affecting Demand:
4. Changes in tastes and preferences.
No exact relationship, depends on the specific changes.
Example: cholesterol scare  demand for eggs ↓.
Example: Dietary information about the benefits of
fish consumption and the health concerns over red meat
consumption
 demand for fish ↑
 demand for red meats ↓
 demand for chicken and turkey ↑
Factors Affecting Demand:
1. Changes in income (for normal goods)
2. Changes in the price of substitutes
3. Changes in the price of complements.
4. Changes in tastes and preferences.
5. Increase in population or the number of consumers.
Generally, if population ↑  demand
for most commodities ↑.
Elasticity
The concept of elasticity is used as a measure of
consumption responsiveness to changes in a particular
variable (e.g., own price, income, or cross prices i.e., prices
of substitutes or complements).
Elasticity
We will concentrate on 3 elasticity concepts:
• own price elasticity of demand
• income elasticity of demand
• cross price elasticity of demand
We will also evaluate point elasticity rather than arc
elasticity.
Elasticity
Why do economists use elasticity and not slope to measure
responsiveness of demand?
Because you will get a different measure of responsiveness
if you simply change the units of measure on either the
vertical or horizontal axis.
For example:
Now simply change the units of measure of
from $/unit to ¢/unit.
Elasticity
Thus, the slope is not a good measure of responsiveness.
Economists prefer using elasticity to measure responsiveness
because elasticity is in
terms.
Elasticity
Let the demand for good
be:
The own price elasticity of demand measures the
responsiveness of consumption of good
in the price of good
, ceteris paribus.
to changes
Elasticity
Why is the own price elasticity negative?
To reflect the downward sloping demand schedule.
Elasticity
the elastic portion of the demand function.
the unitary elastic point on the demand function.
the inelastic portion of the demand function.
Elasticity
Example:
Estimate the own price elasticity of demand at the
point
(this point lies on the
elastic portion of the
demand function)
Example:
Suppose now you wanted to determine the price elasticity at
(this point lies
on the inelastic
portion of the
demand
function)
Factors affecting the price elasticity of demand:
1. Availability of substitutes
(and closeness of substitutes).
More substitutes and closer substitutes 
more elastic demand.
Factors affecting the price elasticity of demand:
1. Availability of substitutes
(and closeness of substitutes).
2. Uses of the product.
More uses of the product or goods 
more elastic demand.
Factors affecting the price elasticity of demand:
1. Availability of substitutes
(and closeness of substitutes).
2. Uses of the product.
3. Share in consumer budgets.
Commodities with larger shares tend to be more
elastic. Pencils are a small portion of consumers’
budgets, so if the price of pencils changes, one
would not expect a large quantity response.
However, items such as automobiles, appliances, etc.
tend to be more elastic.
Factors affecting the price elasticity of demand:
1. Availability of substitutes
(and closeness of substitutes).
2. Uses of the product.
3. Share in consumer budgets.
4. Luxuries vs. necessities
Factors affecting the price elasticity of demand:
4. Luxuries vs. necessities
Demand for necessities tend to be more inelastic than
luxuries. Demand for gasoline, milk, salt, etc. tend to be
inelastic. Demand for large screen TVs, vacations
abroad, etc. tend to be elastic.
Factors affecting the price elasticity of demand:
1. Availability of substitutes
(and closeness of substitutes).
2. Uses of the product.
3. Share in consumer budgets.
4. Luxuries vs. necessities
5. Time period for consumption.
Factors affecting the price elasticity of demand:
5. Time period for consumption.
Over a long period of time, consumers can
either adjust their budgets to a price change in a
particular commodity or find substitutes.
Consequently, the long run elasticity tends to be
more elastic than the short run price elasticity.
Examples:
Gasoline
Housing
-0.40
-0.30
-1.50
-1.88
Other Elasticities:
a. Income elasticity
The income elasticity measures the responsiveness
of good to changes in income, ceteris paribus.
Plotting
and
(income), one traces out the Engel curve.
Other Elasticities:
a. Income elasticity
b. Cross price elasticity
The cross price elasticity of demand measures
the responsiveness of to changes in the price of
other goods ceteris paribus.
Cross Price Elasticity
 consumers switch to z, so the demand for z ↑
Cross Price Elasticity
As
↑ quantity demanded of
Since and
for ↓
↓
are complements ,  demand
Relationship Among Elasticities
There are 3 key relationships among elasticities:
• homogeneity condition
• Slutsky or symmetry condition
• Engel condition
The theory behind these elasticity relationships makes
a certain assumption regarding individual consumer behavior.
Elasticity Matrix
Given n goods and income, we have the following
elasticity matrix:
Elasticity Matrix
Own price elasticities are located on the diagonal.
Cross price elasticities are on the off-diagonal.
Income elasticities are on the last column.
Homogeneity condition
The homogeneity condition states that the
sum of the own and cross price elasticities and
income elasticities for a particular commodity
is zero.
Homogeneity condition
The homogeneity condition stems from Euler’s Theorem
which states that if a function
is homogeneous of degree k then
If
then the function is homogeneous of degree
zero (HD0).
Homogeneity condition
Since
, we can rewrite this as:
Now divide through by
:
This is the homogeneity condition.
Homogeneity condition
The meaning of the homogeneity condition is that the
magnitude of the own price elasticity must be consistent
with the cross price elasticities and income elasticity of
that commodity.
Example
Demand for beef:
Own price elasticity
-0.62
Cross price with pork
0.11
Cross price with lamb
0.01
Cross price with chicken
0.06
All other cross elasticities
-0.01
Income elasticity
0.45
∑=0
Slutsky or Symmetry Condition
This condition specifies a specific relationship
between
and
The Hotelling-Jureen relationship states:
Hotelling-Jureen
Engel Condition
The consumer’s budget can be written as:
assuming all income is spent on
commodities
The effects of changes in income on consumption can
be obtained by differentiating the above equation
with respect to I :
Engel Condition
Multiply each component by
Engel Condition
Recall that:
and
This states that the weighted sum of the income elasticities
for all items in the consumer’s budget should sum to 1.
Market Demand Conditions
How are market demand functions obtained from
consumer demand functions?
Quantities demanded or purchased by consumers are
added together for each price level.
Market Demand Conditions
Market Demand Conditions
Consumer #1
Consumer #2
Consumer #3
Market Demand Conditions