ECON 7030 Microeconomic Analysis
Lecture 6: Demand
University of Queensland
Semester 1, 2022
Last year. . .
• We have seen how we can find utility maximising and
expenditure minimising bundles
• You may think: Let’s practise calculating the optimal bundle!
• That’s not a bad idea for assessments, but is of limited use in
economic studies
Finding Optimal Bundles is Just the Beginning
• Utility is a theoretical construct, it is not observable
• In applications, we can rarely get the “real” utility function of
an individual and calculate his/her optimal consumption
bundle
• Also, since utility is not observable, if I am looking at a single
choice, I can always come up with an ad-hoc utility function
that makes that choice optimal
Goal: Testable Predictions
• The purpose of utility maximisation and expenditure
minimisation is to be able to give predictions on consumers’
behaviours
• We also want these predictions to be testable — that is,
empirical observations or data can (at least potentially)
differentiate our predictions against alternative predictions.
• One way to do it is to look at how choices change when
observable variables change
• This is also useful for applications, as while we may not have
a “real” utility function, we may still predict the effect of
certain policies
What Are We After?
• Recall that the condition for a utility maximising bundle is
given by
Ux(x , y)
Uy (x , y)
=
Px
Py
Pxx + Pyy = M
• There are two kinds of variables
Endogenous Variables (Choice Variables): x , y
Exogenous Variables (Parameters): Px , Py , M
• What we want: To be able to say how the endogenous
variables change when an exogenous variable change
(This is known as Comparative Statics)
Utility Maximising Bundle as a Function of the
Parameters
• We have two equations in two unknowns:
Ux(x
∗, y∗)
Uy (x∗, y∗)
=
Px
Py
Pxx
∗ + Pyy∗ = M
• Unless we are really unlucky, a solution exists — we will call it
(x∗, y∗)
• Notice that x∗ and y∗ depends on Px , Py and M — when the
value(s) of Px , Py and M change(s), the values of x
∗ and y∗
change
• In other words, x∗ and y∗ are functions of Px , Py and M
(Marshallian) Demand
• We have
x∗ = x∗(Px ,Py ,M)
y∗ = y∗(Px ,Py ,M)
• These functions express the quantity of x and y the consumer
is willing and able to purchase given the prices and income
• Or, these function express the quantity demanded for Good X
and Good Y given the prices and income
• In other words, the functions x∗ and y∗ are our (Marshallian)
demand functions
Comparative Statics
x∗ = xm(Px ,Py ,M)
y∗ = ym(Px ,Py ,M)
• There are three changes (in x∗) that we are interested in
Own Price Effect Change in x∗ due to a change in Px
Cross Price Effect Change in x∗ due to a change in Py
Income Effect Change in x∗ due to a change in M
• Associated with these effects are three elasticities:
Own Price Elasticity % change in x∗ / % change in Px
Cross Price Elasticity % change in x∗ / % change in Py
Income Elasticity % change in x∗ / % change in M
• We are more interested in elasticities than just changes
because elasticity is unit free — even if you change the unit of
measurements, you are not going to change the elasticity
Elasticity: In General
• Suppose I have two variables, v and w , and suppose
v = v(w , α), where α is some other parameters
• The elasticity of v with respect to w is
% change in v
% change in w
=
∆v
v × 100
∆w
w × 100
=
w
v
∆v
∆w
=
w
v
∂v
∂w
A Little Trick for Those Interested
• If both v and w are always positive,
w
v
∂v
∂w
=
∂ ln v
∂ lnw
• Given the Cobb-Douglas Utility, U(x , y) = xαy1−α
We can solve for the utility-maximising x∗ and get
x∗ = α
M
Px
Hence
ln x∗ = lnα + lnM − lnPx
• Which gives
Own Price Elasticity =
∂ ln x∗
∂ lnPx
= −1
Income Elasticity =
∂ ln x∗
∂ lnM
= 1
Digression: A Little Trick for Econometric Estimations
• Due to the little trick, when you run your regression in
econometrics, you can estimate
ln x∗ = β0 +β1 lnPx +β2 lnPy +β3 lnM + other controls, etc.
• Then β1, β2 and β3 will give you the own-price elasticity,
cross-price elasticity and income elasticity, respectively
Rest of This Lecture
1 Change in Income
2 Change in Price
3 Featured Example
Part I
Change in Income
Change in Income
x
y
U0
M
M ′
xA
yA A
B
C
D
• Question: Where should the new bundle be? Like B, C , or D?
Answer:
Both Goods are Normal
x
y
U0
U1
M
M ′
xA
yA
xC
yC
A
C
Both X and Y are normal goods
• The consumption of a normal good increases when income
increases
X is Inferior and Y is Normal
x
y
U0
U1
M
M ′
xA
yA
xB
yB
A
B
X is an inferior good
Y is a normal good
• The consumption of an inferior good decreases when income
increases
X is Normal and Y is Inferior
x
y
U0
U1
M
M ′
xA
yA
xD
yD
A
D
X is a normal good
Y is an inferior good
Can Both Goods Be Inferior?
x
y
U0
M
M ′
xA
yA
xE
yE
A
E
• If consumption of both goods falls, the consumer will not
exhaust his/her budget
Budget Balancedness
• In fact, the prediction that the consumer will exhaust his/her
budget is one that we can test
• We know
Pxx
∗(Px ,Py ,M) + Pyy∗(Px ,Py ,M) = M
When there is a change in M,
Pxx
∗
M
M
x∗
∆x∗
∆M
+
Pyy
∗
M
M
y∗
∆y∗
∆M
= 1
sxηx + syηy = 1
where sx and sy are the expenditure shares of x and y ,
respectively
and ηx and ηy are the income elasticities of x and y ,
respectively
Budget Balancedness
• The equation
sxηx + syηy = 1
says that the weighted average (by expenditure shares) of
income elasticities of all goods has to be 1
• This equation extends to more than 2 goods as well
• This is something that we can test with empirical data
• Also this rules out some possibilities
Some Terminology
Income Elasticity Category Sub-category
Negative Inferior Good
Positive, < 1
Normal Good
Necessity
Positive, ≥ 1 Luxury
• Since the weighted average of income elasticities has to be 1,
there must be at least one luxury good
Engel Curve
• If we trace the path of consumption bundles as income
changes, we get an Engel Curve
0
x
y
Engel CurveM1
U1
xA
yA
M2
U2
xB
yB
M3
U3
xC
yC
Engel Curve can bend
• A good may be normal at some income range but inferior at
another
0
x
y
Engel Curve
M1
U1
xA
yA
M2
U2
xB
yB
M3
U3
xC
yC
Going from Engel Curve to Demand Curves
0
x
y
Engel Curve
xA
yA
xB
yB
xC
yC
0
x
Px
Px
D1 D2 D3
When Income Increases: Summary
Normal Goods Inferior Goods
η Positive
Negative
Demand shifts. . . Right
Left
Engel Curve slope upward* Downward**
*Both goods are normal.
**One good is inferior and another good is normal.
Part II
Change in Price
Change in Price
0
x
y
M1
slope=− px
py
U1
A
xA
yA
slope=− p′x
py
Px falls, Py unchanged
Two Effects of a Change in Price
0
x
y
M1
slope=− px
py
slope=− p′x
py
When price of x falls,
1 Relative price of x to y falls
(budget line becomes flatter)
∗ Substitution Effect
2 Real income increases
(more bundles are affordable)
∗ Income Effect
Substitution Effect
• To find the substitution effect, we change the relative price
while holding real income constant
• But what do we mean by “real income constant”?
• We will consider a hypothetical income such that the
consumer can achieve just the original utility level
• This is known as the Hicksian decomposition method
• That’s expenditure minimisation!
Substitution Effect: Hicksian Method
x
y To find the substitution effect, we
change the relative price while hold-
ing utility constant
M1
slope=− Px
Py
U1
A
slope=− P′x
Py
slope=− P′x
Py
H
xA xH
A → H: Substitution Effect
Substitution Effect is Always Negative
(i.e., when Px falls, x
∗ increases)
x
y
When relative price of x falls, budget line
becomes flatter. When the new budget line
is tangent to the old indifference curve, the
slope of the indifference curve is also flatter.
By diminishing MRS, this new tangent point
must occur with a larger quantity of x .M1
U1
A
slope=− P′x
Py
slope=− P′x
Py
H
xA xH
A → H: Substitution Effect
Finding the Substitution Effect
Steps:
1 To find the substitution effect algebraically, first calculate the
utility level at the original bundle (A)
2 Then run an expenditure minimisation with the new prices
and the original utility level
3 The expenditure minimising bundle is your bundle H
4 The change in quantities between bundle A and H is the
substitution effect
Substitution Effect and Expenditure Minimisation
• If you remember what we were talking about duality last
week, you might note that if I had run an expenditure
minimisation with the old prices and the original utility level, I
will get bundle A
• Just now I said I will get bundle H if I run expenditure
minimisation with the new prices and the original utility level,
I will get bundle H
• In other words, the substitution effect is the change in the
expenditure minimising bundle when I change the price (of x)
holding utility level constant
Hicksian (Compensated) Demand
• Recall that we can express our expenditure minimising bundle
(given a utility level) as
xh = xh(Px ,Py ,U)
yh = yh(Px ,Py ,U)
• Like the Marshallian Demands these functions express the
quantity demanded for x and y given the prices and utility
level
• This type of demand, with utility level (instead of income)
fixed, is known as Hicksian Demand or Compensated Demand
• As Hicksian demand considers only the substitution effect, it
is always downward-sloping
Going from Indifference Curves to Hicksian Demand
x
y
M1
U1
A
slope=− P′x
Py
slope=− P′x
Py H
xA xH
x
Px
Px
P ′x
Dh
Income Effect
x
y To find the income effect, we
change real income while holding
relative price constant
M1
U1
A
slope=− P′x
Py
slope=− P′x
Py
H
xA xH
Income Effect: Normal Goods
x
y Consumption of a Normal Good in-
creases as real income increases
M1
U1
A
slope=− P′x
Py
slope=− P′x
Py
H
xA xH
U2
B
xB
A → H Substitution Effect
H → B Income Effect
A → B Price Effect
Income Effect: Inferior Goods
0
x
y Consumption of an Inferior Good
falls as real income increases
M1
U1
A
slope=− P′x
Py
slope=− P′x
Py
H
xA xH
U2
B
xB
A → H Substitution Effect
H → B Income Effect
A → B Price Effect
Income Effect: Giffen Goods
x
y Income Effect of Giffen Good out-
weighs Substitution Effect
M1
U1
A
slope=− P′x
Py
slope=− P′x
Py
H
xA xH
B
xB
A → H Substitution Effect
H → B Income Effect
A → B Price Effect
Price Effect for Different Kinds of Goods
• When Price of X falls,
Normal Good Inferior Good Giffen Good
Substitution Effect QX rises QX rises QX rises
Income Effect QX rises QX falls QX falls
Price Effect QX rises QX rises QX falls
Marshallian (Uncompensated) Demand
• If we consider the full price effect, we can plot the quantity of
x demanded given a fixed income level (and holding Py
constant) by varying Px
• In other words, we are looking at a function
xm = xm(Px ,Py ,M)
(which is the quantity coming from utility maximisation)
• This type of demand is known as Marshallian Demand or
Uncompensated Demand
• Since Giffen good is theoretically possible, a Marshallian
Demand curve can be upward sloping
Going from Indifference Curves to Marshallian Demand
x
y
M1
U1
xA
A
slope=− P′x
Py
slope=− P′x
Py
xH
H
U2
xB
B
x
Px
Px
P ′x DmDh
Marshallian and Hicksian Demand
• You may notice that the Marshallian Demand is flatter than
the Hicksian Demand in the last slide
• That is because we are drawing the case of a normal good,
where the substitution effect and the income effect reinforces
each other
• In general:
Normal Good Marshallian Demand is flatter
Inferior Good Marshallian Demand is steeper (and is positive
if it is a Giffen Good)
Marshallian and Hicksian Demand Elasticities
• Recall the Duality result from last lecture:
xh(Px ,Py ,U) ≡ xm(Px ,Py ,E ∗(Px ,Py ,U))
• When there is a change in Px , then
Px
x
∆xh
∆Px
=
Px
x
∆xm
∆Px
+
Pxx
M
M
x
∆xm
∆M
εhxx = ε
m
xx + sxηx
εmxx = ε
h
xx − sxηx
where εhxx and ε
m
xx are the own price elasticities of x according
to the Hicksian and the Marshallian demand, respectively
Slutsky Equation
• The equation
εmxx︸︷︷︸
Price Effect
= εhxx︸︷︷︸
Sub. Effect
−sxηx︸ ︷︷ ︸
Income Effect
is known as the Slutsky Equation
• It decomposes the Substitution Effect and Income Effect
algebraically
• Moreover, it says that the size of the income effect is
proportional to the expenditure share of Good X
• This explains why it is so hard to find an example of a Giffen
Good: to be strongly inferior, the good tends to be narrowly
defined, which means it tends to have a small expenditure
share
Category ηx εmxx Sub-category
Inferior Good ηx < 0
εmxx > 0 Giffen Good
εmxx < 0
Normal Good
0 ≤ ηx < 1 εmxx < 0
Necessity
1 ≤ ηx Luxury
εmxx = ε
h
xx − sxηx
Part III
Featured Example
Featured Example: Lightbulb
• Suppose you buy lightbulbs only to get light (i.e., lightbulb
brings no utility per se)
• And lightbulb is the only input for getting light
• Due to a technological innovation, all lightbulbs now last
twice as long
• The price of lightbulb remains unchanged
• Would you spend more, less or the same amount on lightbulb
now?
Light and Lightbulb
• If you want the same amount of light, your demand for
lightbulbs will be halved
• But the effective price of each unit of light is also halved —
and you demand more light due to the price effect
• If you demand
• exactly twice as much light as before =⇒ demand for lightbulb
unchanged
• more than twice as much light as before =⇒ demand for
lightbulb increase
• less than twice as much light as before =⇒ demand for
lightbulb decrease
• What tells us which case it is?
Our Lightbulb Moment
• It is the
price elasticity of demand for light!
Demand for Light Demand for Lightbulb Expenditure on Lightbulb
Inelastic
Decrease Decrease
Unitarily Elastic
Unchanged Unchanged
Elastic
Increase Increase
Other Applications
Environmental Economics
Cars become more energy efficient and consume less
gasoline per km driven. Would the demand for
gasoline increase, decrease or remain unchanged?
Labour Employment
Increase in capital means that each worker can now
produce the output of 2 workers before. Would
labour demand increase, decrease or remain
unchanged?
Quality Improvement
Technological advancement allows a small phone in
2010 to have more computation power than a
mainframe in 1970. Do people buy more, fewer or
the same quantity of computers (broadly defined)?
Summary
• The effect of change in income on demand
• The effect of change in prices on demand
• Income Effect
• Substitution Effect
• Slutsky Equation
• Featured Example: Elasticity and derived demand