Lecture: Demand
Demand
Uncompensated Demand
By de…nition demand tells us how much a consumer is willing to buy
at a given price, holding constant other factors (such as tastes and
preferences, income, and prices of complements and substitutes).
Take the utility maximization problem analyzed earlier.
We solved for the optimal quantities as functions of prices and income.
That is, we have found the consumer’s uncompensated demand
functions for these goods:
q1
q2
= Z ( p1 , p2 , Y )
= B ( p1 , p2 , Y ).
In the Cobb-Douglas example, the uncompensated demands were
q1 = αY/p1 ,
q2 = (1
α)Y/p2 .
The two goods are neither complements nor substitutes: the demand
depends only on the good’s own price.
Demand
Uncompensated Demand (Continued)
Suppose that we double all prices and income. What will happen to
uncompensated demand?
The utility maximization problem becomes
max U (q1 , q2 )
q1 ,q2
s.t. 2p1 q1 + 2p2 q2 = 2Y.
Note that the new budget constraint is equivalent to the original one.
So the two problems will have identical solutions.
If we multiply all prices and income by some number, uncompensated
demand will not change!
Consider our Cobb-Douglas example:
q1 =
α2Y
αY
=
,
2p1
p1
q2 =
(1
Demand
α)2Y
( 1 α )Y
=
.
2p2
p2
Constructing the Uncompensated Demand Curve
Now we use a graphical approach to construct the uncompensated
demand curve.
If we increase a good’s price while holding other prices, tastes and
income constant, the budget constraint will rotate.
Thus, the optimal consumption bundle will change as the budget
constraint rotates.
If we connect all optimal bundles in this graph, we will get the price
consumption curve.
From the price consumption curve we can obtain the demand curve.
Demand
Constructing the Uncompensated Demand Curve (Cont.)
On the graph below, we vary the price of good 1 (beer)
Connecting the equilibrium bundles e1 , e2 , e3 yields the price
consumption curve.
Demand
Constructing the Uncompensated Demand Curve (Cont.)
We can translate the price consumption curve from a (q1 , q2 )-diagram
to a (q1 , p1 )-diagram. This gives us the demand for beer!
Demand
Constructing the Uncompensated Demand Curve (Cont.)
For some preferences the demand curve can be upward sloping!
Good y
Such goods are called Gi¤en goods.
Price consumption
curve
U3
U2
U1
I
Px1
Price of x
P1
Px falls
I
Px 2
I
Px 3
Good x
P2
P3
Demand curve
x
x3 x2 x1
Demand
How Income Changes Shift Uncompensated Demand
In an earlier lecture, we argued that a change in income causes a shift
of the (uncompensated) demand curve.
Now we examine how consumer behavior changes when income
changes, while keeping prices and tastes constant.
A rise in income leads to a parallel outward shift of the budget line.
Thus, as income changes, the optimal bundle will change.
If we connect the optimal bundles for all income levels, we get the
income-consumption curve.
Demand
How Income Changes Shift Demand (Continued)
On the graph below, we vary the consumer’s income.
Connecting the equilibrium bundles e1 , e2 , e3 gives us the
income-consumption curve.
Demand
How Income Changes Shift Demand (Continued)
We can plot the optimal quantity of beer in (q1 , p1 )-space.
The change in income shifts the demand curve.
Demand
How Income Changes Shift Demand (Continued)
We can plot the income-consumption curve in a (q1 , Y )-diagram.
This will give us the Engel curve.
Demand
How Income Changes Shift Demand (Continued)
It is possible that the income-consumption curve is downward-sloping,
and hence the Engel curve could be downward-sloping.
y
Income
consumption
curve
D
U4
C
B
U3
U2
A
BL2
Weekly income I
BL1
I4
I3
I2
I1
BL3
U1
BL4
x
D”
C”
B”
A”
Engel Curve
x
Demand
Consumer Theory and Income Elasticities
Remember the formula for income elasticity of demand:
ξ=
∂Q Y
∆Q/Q
=
.
∆Y/Y
∂Y Q
Some goods have negative income elasticities: ξ < 0.
As income increases, we consume less of them.
They are called inferior goods.
Example of inferior goods: potatoes.
As a person’s income rises, she tends to consume less of them.
Potatoes are replaced with more nutritious food.
If ξ > 0, the good is called a normal good.
If ξ > 1, the good is called a luxury good.
If 1 > ξ > 0, the good is called a necessity.
Demand
Income-Consumption Curve and Income Elasticities
If the income-consumption curve is upward-sloping, all goods are
normal.
If the income-consumption curve is downward-sloping, one good is
normal and the other good is inferior.
Demand
Compensated Demand
Alternatively, we could derive compensated demand functions.
Compensated demand shows how the quantity demanded changes as
the price rises, holding the utility constant.
In contrast with the uncompensated demand, here we vary the
consumer’s "real" income, so he can maintain a constant level of
utility as the price changes.
Hence the term compensated (sometimes called Hicksian) demand.
The compensated demand system is the solution to the expenditure
minimization problem. Its form is
q1 = H ( p1 , p2 , Ū )
q2 = M ( p1 , p2 , Ū ).
In our Cobb-Douglas example, the compensated demand is
q2 = Ū
(1
α ) p1
αp2
α
,
q1 = Ū
Demand
αp2
(1 α ) p1
1 α
.
Deriving Compensated Demand Graphically
Suppose that we vary p x .
To maintain the same level of utility, we must also vary Y accordingly!
Plot the curve in a ( x, p x )-diagram to get compensated demand.
px
A
px
B
px ’
Compensated
Demand
x1
x2
Demand
x
E¤ects of a Price Increase
Consider an increase in the price of good 1.
This has two e¤ects on the uncompensated demand of good 1.
substitution e¤ect: the change in the quantity of a good that a
consumer demands when the good’s price rises, holding other prices
and the consumer’s utility constant.
income e¤ect: the change in the quantity of a good that a consumer
demands due to the change in his "real" income, holding prices …xed.
The total e¤ect on uncompensated demand is the sum of the income
and the substitution e¤ects.
The substitution e¤ect always goes in one direction: buy more of the
good that becomes cheaper.
The income e¤ect can go either way, depending on whether the good
is normal or inferior.
Demand
Decomposing the E¤ects Graphically
Suppose that the price of good 1 has increased.
Thus, the budget constraint rotates inward around the vertical axis.
We decompose the total e¤ect as follows.
1
Find the optimal bundle before the price change.
2
Find the optimal bundle after the price change.
3
Now draw an "imaginary" budget line which is parallel to the new
budget line, but tangent to the old indi¤erence curve.
The change between the two parallel lines is the income e¤ect.
The change due to rotating the budget line around the old
indi¤erence curve is the substitution e¤ect.
Demand
Decomposing the E¤ects Graphically (Continued)
The e¤ects are illustrated on the graph below:
Demand
Numerical example
0.4
Suppose that the consumer’s utility is U (q1 , q2 ) = q0.6
1 q2 .
We know that the corresponding uncompensated demands are
q1 = 0.6
Y
,
p1
q2 = 0.4
Y
.
p2
Assume that p1 = 15, p2 = 20, Y = 300. Then the price of good 1
increases to p10 = 30.
Initially the consumer buys 0.6 300/15 = 12 units of good 1 and
0.4 300/20 = 6 units of good 2.
After the price increase she buys 0.6 300/30 = 6 units of good 1
and 6 units of good 2.
Her consumption of good 1 decreases by 6 units (from 12 to 6).
How much of this change is due to the income e¤ect and how much
due to the substitution e¤ect?
Demand
Numerical example
Before the price increase, the consumer’s utility was Ū = 120.6 60.4 .
After the price increase, how much income Y does the consumer
need to have to attain the old level of utility Ū?
This income would solve the equation
0.6Y
30
0.6
0.4Y
20
0.4
= 120.6 60.4
Solving for Y yields Y = 450.
If the consumer had income Y = 450, at the new price p10 he would
buy 0.6 450/30 = 9 units of good 1.
The decrease from 12 to 9 is due to the substitution e¤ect.
We have maintained the "real" income constant, so the consumer’s
level of utility is unchanged.
The remaining change is due to the income e¤ect.
Demand
Price Changes and Normal Goods
Suppose the price of a normal good increases (as in slide 19).
This good would become relatively more expensive.
Thus, the substitution e¤ect says that we should consume less of it.
Furthermore, the consumer’s real income decreases (the old
consumption bundle becomes una¤ordable).
Since the good is normal, the income e¤ects says that we should
consume less of it.
Thus, both e¤ects work in one direction: consume less.
The demand curve of a normal good is necessarily downward-sloping!
Demand
Price Changes and Inferior Goods
Next suppose that the price of an inferior good increases.
Then this good would become relatively more expensive.
Thus, the substitution e¤ect says that we should consume less of it.
Furthermore, the consumer’s real income decreases (the old
consumption bundle becomes una¤ordable).
Since the good is inferior, the income e¤ects says that we should
consume more of it.
The two e¤ects work in di¤erent direction!
The overall e¤ect depends on which e¤ect is stronger.
Demand
Price Changes and Inferior Goods (Continued)
Suppose that the substitution e¤ect is stronger.
Then demand will still be downward-sloping!
y
B
A
U1
Income effect
C
BLd
BL1
BL2
U2
x
xB xC xA
Demand
Price Changes and Inferior Goods (Continued)
What if the income e¤ect is stronger?
Then we have a Gi¤en good: demand is upward-sloping!
All Gi¤en goods are inferior. But the converse is not true!
y
U1
U2
x
Income effect
Demand
Changes in Compensated versus Uncompensated Demands
Note that the substitution e¤ect is actually the change in
compensated demand we get if we …x the utility at the old level!
If we set the consumer’s income to be equal to the minimized
expenditure, the compensated demand will be the same as the
uncompensated demand.
Next consider a decrease in the price of good 1.
If the good is normal, the income e¤ect will increase consumption.
Therefore, the compensated demand will increase by less than the
uncompensated (will be steeper).
If the good is inferior, the income e¤ect will decrease consumption.
The compensated demand will increase by more than the
uncompensated (will be ‡atter).
Demand
Compensated vs Uncompensated Demands (Continued)
Illustration of a normal good:
Demand
The Slutsky Equation
The relationship between the total change in demand, the income and
the substitution e¤ects can be demonstrated mathematically.
This relationship is known as the Slutsky equation:
∂D
∂H
=
∂p1
∂p1
∂D
∂p1 is the
∂H
∂p1 is the
∂D
∂Y q1 is
∂D
q1 .
∂Y
change in the uncompensated demand (the total change).
change in the compensated demand (the substitution e¤ect)
the income e¤ect.
We can rewrite the Slutsky equation as
ε=ε
θξ,
where ε is the price elasticity of demand, ε is the elasticity of
compensated demand, ξ is the income elasticity of demand, and
θ = p1 q1 /Y is the expenditure share.
Demand
Remember the numerical example with
0.4
0
U (q1 , q2 ) = q0.6
1 q2 , p1 = 15, p1 = 30, p2 = 20, Y = 300.
The uncompensated demands are q1 = 0.6 pY1 , q1 = 0.4 pY2 .
The elasticity of uncompensated demand for good 1 is
ε=
∂q1 p1
=
∂p1 q1
0.6
Y p1 p1
=
p21 0.6 Y
1.
The income elasticity of good 1 is
ξ=
∂q1 Y
1 Y p1
= 0.6
= 1.
∂Y q1
p1 0.6 Y
The compensated demands are q1 = Ū
3p2
2p1
0.4
, q1 = Ū
2p1
3p2
0.4
.
The elasticity of compensated demand for good 1 is
ε=
∂q1 p1
=
∂p1 q1
The market share is θ =
0.4Ū
15 12
300
3p2
2p1
0.6
3p2 p1
2p21
2p1
3p2
0.4
=
0.4.
= 0.6. The Slutsky equation holds!
Demand