Intermediate microeconomics Lecture 2: Consumer Theory II: demand. Varian, chapters 6, 8, 9 Agenda
1. Normal and inferior goods
2. Income offer curves and Engel curves
3. Ordinary goods and Giffen goods
4. Price offer curves and demand curves
5. Substitutes and complements
6. The Slutsky equation 7. The Law of Demand
8. The Slutsky equation with enowments
2017-01-21 Adam Jacobsson, Department of Economics 2 1. Normal and inferior goods
Definition 1 (normal good). A good is normal if the demanded quantity increases if non-labour income increases: �� > 0. �� Definition 2 (inferior good). A good is inferior if the demanded quantity decreases if non-labour income increases: �� < 0. ��
2017-01-21 Adam Jacobsson, Department of Economics 3 Example: Normal good: m increases: � < �′
Good 2 �′
�2
�
�2
Good 1 �1 �′1 � �′
�1 �1 2017-01-21 Adam Jacobsson, Department of Economics 4 Example: Inferior good: m increases: � < �′
Good 2 �′
�2
�
�2
Good 1 �′1 �1 � �′
�1 �1 2017-01-21 Adam Jacobsson, Department of Economics 5 Good 2 Income offer �′
�2 curve
�
�2 Income offer and Engel curves � > �
Good 1 �1 �′1 � �′ m �1 �1
Engel curve
�′
�
Good 1 �1 �′1
2017-01-21 Adam Jacobsson, Department of Economics 6 3. Ordinary and Giffen goods
Definition 3 (Ordinary good). A good is ordinary if the demanded quantity decreases when the price of the good increases: �� < 0. ��� Definition 4 (Giffen good). A good is a Giffen good if the demanded quantity increases when the price of the good increases: �� > 0 ���
2017-01-21 Adam Jacobsson, Department of Economics 7 Example: Ordinary good: �1 increases: �1 < � 1
Good 2 �
�2
� � Good 1 �′1 �1 � 1 �1
2017-01-21 Adam Jacobsson, Department of Economics 8 Example: Giffen good: �1 increases: �1 < � 1
Good 2 �
�2
� � Good 1 �1 �′1 � 1 �1
2017-01-21 Adam Jacobsson, Department of Economics 9 Good 2 Price offer �′
�2 curve
Price offer and Demand curves �1 < � 1
Good 1 �′1 �1 � � �′ �1 �1
Demand curve � 1
�1
Good 1 �′1 �1
2017-01-21 Adam Jacobsson, Department of Economics 10 5. Substitutes and Complements
Definition 5. Good 1 is a substitute for good 2 if
�� > 0. ��2 Definition 6. Good 1 is a complement for good 2 if
�� < 0. ��2
2017-01-21 Adam Jacobsson, Department of Economics 11 6. The Slutsky equation
A change in the price of a good has two effects:
1. The relative price of the good is affected: – Substitution effect 2. The purchasing power is affected: – Income effect
2017-01-21 Adam Jacobsson, Department of Economics 12 Deriving the Slutsky equation, method 1: (�1 decreases: �1 > �′1 )
Good 2 x1A(p1, p2, m) � x1B(p’1, p2, m’) �2 x1C(p’1, p2, m)
SE = x1B(p’1, p2, m’) - x1A(p1, p2, m) IE = x (p’ , p , m) - x (p’ , p , m’) C 1C 1 2 1B 1 2
A B
SE IE Good 1 �1� �1� � �1� �′ �
�1 �′1 �′1 2017-01-21 Adam Jacobsson, Department of Economics 13 The total change in demanded quantity is SE+IE!
Δx1= x1B(p’1, p2, m’)− x1A(p1, p2, m) + x1C(p’1, p2, m) − x1B(p’1, p2, m’) s n x 1 x 1
m n We define Δx 1=-Δx 1. We then get: s m Δx1= Δx 1 - Δx 1 (1)
Divide both sides by Δp1 to get the “rates of change”:
x xs xm 1 = 1 − 1 (2) p1 p1 p1
2017-01-21 Adam Jacobsson, Department of Economics 14 For m and m’ we have: � = � � + � � 1 1 2 2 (3) �′ = � 1�1 + �2�2
Where the budget lines intersect (point A) we have: �� = � − � = �1 � 1 − �1 = �1Δ�1 (4) or Δ�1 = (5) Equations (5) and (2) give us the Slutsky equation
2017-01-21 Adam Jacobsson, Department of Economics 15 The Slutsky equation Δ� Δ�1 = �1
Use (5) in (2) to rewrite the income effect: m m x 1 x 1 First let us rewrite the income effect: = �1 p1 m (2) Then becomes:
s m x1 x 1 x 1 = − �1 (6) p1 p1 m . .
2017-01-21 Adam Jacobsson, Department of Economics 16 Example: normal good and p1 increases
xs 1 <0 a negative substitution effect & p1
xm x 1 >0 a positive income effect yields: 1 m s m Δx1 Δx 1 Δx 1 = − �1 <0 Δ p1 Δ p1 Δm . .
Let us work through an example!
2017-01-21 Adam Jacobsson, Department of Economics 17 Deriving the Slutsky equation, method 2: (�1 decreases: �1 > �′1 )
Now we keep the utility constant instead of purchasing power.
Good 2 � The Hicksian substitution �2 effect …is virtually the same as the Slutsky subst effect for �(� , � , � ) 1 2 small changes in p1. C �2 A �′ B
� SE IE Good 1 � �(� 1, �2, � ) �
�1 �′1 �′1 2017-01-21 Adam Jacobsson, Department of Economics 18 7. The law of demand
If demand for a good increases when non-labour income increases, then demand for the same good will decrease if the price of the good increases. That is: If a good is normal, then it must be ordinary.
s m x1 x 1 x 1 Proof: = − �1 <0 p1 p1 m ( ) ( ) ( ) The opposite need not be true!
2017-01-21 Adam Jacobsson, Department of Economics 19 8. The Slutsky equation with endowments
● Now assume that individuals’ income (m) is composed of endowments
of the two goods:
� = �1�1 + �2�2
● Where �1 is the endowment of good 1 and �2 is the endowment of good 2.
– (�1 could be the amount of apples and �2 the amount of oranges that a farmer produces each year)
● A change in, for example �1, will have the following effects: – a substitution effect (as before), – an ordinary income effect (as before), and – an endowment income effect.
2017-01-21 Adam Jacobsson, Department of Economics 20 ● Assume �1 increases to �′1 which means the value of the endowment (income) is now: Can be re-written �′′ = �′ � + � � 1 1 2 2 as: Δ� = 1 This means that: �� = (� 1�1 + �2�2) − (�1�1 + �2�2) = � − � = Δ�1�1 Proceeding as we did with the first Slutsky equation:
x (p′ , p , m′′)−x (p , p , m) 1 1 2 1 1 2 = Δ�1 x (p , p , m )−x (p , p , m) 1 1 2 1 1 2 (Marshallian substitution effect) x (p , p , m )−x (p , p , m) − 1 1 2 1 1 2 (ordinary income effect) x1 (p 1, p2, m )−x1 (p 1, p2, m) + (endowment income effect) ● By definition of the ordinary income effect, see equation (5): � − � Δ�1 = �1 ● … and by definition of the endowment income effect: Δ�1 = , we can rewrite the new Slutsky equation:
x (p’ , p , m′′)−x (p , p , m) x (p′ , p , m′)−x (p , p , m) 1 1 2 1 1 2 = 1 1 2 1 1 2 Δ�1 Δ�1
x (p′ , p , m′)−x (p′ , p , m) − 1 1 2 1 1 2 � � − � 1
x (p′ , p , m′′)−x (p′ , p , m) + 1 1 2 1 1 2 � � − � 1
2017-01-21 Adam Jacobsson, Department of Economics 22 ● Re-writing using ∆�:
s m � Δx1 Δx 1 Δx 1 Δx 1 = − �1 + �1 Δ p1 Δ p1 Δm Δm . . . . .
● For small changes i price: ≈ ⟹
s m Δx1 Δx 1 Δx 1 = + (�1 − �1) Δ p1 Δ p1 Δm . .
2017-01-21 Adam Jacobsson, Department of Economics 23