ECON2913 (Spring 2012)
6 & 9.3.2012 (Tutorial 4)
Income and substitution effect
X2




A
X
2X2’
E
N
M
IC’
P1
P2
IC
X1
B
Initial equilibrium: E with P1, P2 and Y
Suppose P1 to P1’, P2 and Y remains unchanged.
New equilibrium: N with P1’, P2 and Y
The price effect is composed of two effects:
substitution effect (SE) and income effect (IE).
’
X1S
1
X1
P1'
P2
X1
C
IE
SE
Substitution effect: change in consumption of a good associated with a change in its price,
with the level of utility held constant.
 SE is always negative, i.e. when P1  X1 and vice versa.
Income effect: change in consumption of a good resulting from an increase in purchasing
power, with relative prices held constant.
 IE can be positive or negative
 Normal goods: positive IE, i.e. Y X1
 Inferior goods: negative IE, i.e. Y X1
Derivation of demand curve (Uncompensated Demand)
X2
A
Y
(IE)Y
X (IE)X
Z (IE)Z
E
X2
N
’
X2
M
IC’
P1
P2
X1
SE
P1 
IE
SE
IC
B
X1S
(IE)N
X1
’
1
P1'
P2
X1
C
 X1
Normal good (+IE)  X1  IE + SE  X1 (Point N)
Inferior good (IE)  X1
(a) SE > IE (X1 > X1)  IE + SE  X1 (Point Z)
Downward sloping DD
Downward sloping DD
(b) SE = IE (X1 = X1)  IE + SE  X 1 (Point X)
Vertical DD
(c) SE < IE (X1 < X1)  IE + SE  X1 (Point Y)
Upward sloping DD
(Giffen goods)
1
Compensated and Uncompensated demand curve
 Compensated (Hicksian)demand curve: x = h(px, py, U)
 It shows the quantity demanded for a good at each price, holding utility constant.
(Substitution effect only, real income remains constant)
 Uncompensated (Marshallian, Ordinary) demand curve: x =l(px, py, I)
 It shows the quantity demand for a good at each price, holding nominal income constant.
(Substitution effect and Real income effect)
X2
A
Y
(IE)Y
X (IE)X
Z (IE)Z
E
X2
N
’
X2
M
IC’
P1
P2
X1
SE
P1'
P2
IC
’
X1S
B
(IE)N
X1
1
X1
C
P1
Compensated DD (SE only)
E
P1
P1’
M
 Good X1 is a normal good
 When P drops, real income is higher
and X1 is higher
 Real income effect is positive
 Normal good
Uncompensated DD
 More elastic uncompensated DD
(SE + IE)
N
X1
X1S
X1
X1’
P1
Uncompensated DD
(SE + IE)
P1
 Good X1 is an inferior good
 When P drops, real income is higher
and X1 is lower
 Real income effect is negative
 Inferior good
 More inelastic uncompensated DD
E
P1’
Compensated DD (SE only)
Z
M
X1
X1
X1S
X1S
2
Demand relationships among goods and Elasticities
(1) Own price effect

Normal good if

Giffen good if

Price elasticity of demand
(downward sloping demand)
(upward sloping demand)
(2) Income effect

Normal good if
(downward sloping demand)

Inferior good if
(downward/ upward sloping demand)

Income elasticity of demand


Luxury/ superior good if
Necessity if
(3) Cross price effect

Gross substitutes if

Gross complement if

Does

Cross-price elasticity of demand
imply
?
Elasticities and Demand functions

Does the own price/ income/ cross price elasticity of demand depends on px, I and py?
(1) Linear demand function:

Price elasticity of demand

Income elasticity of demand

Cross-price elasticity of demand
(2) Non-linear demand function:

Price elasticity of demand

Income elasticity of demand

Cross-price elasticity of demand
(
and
3
(3) Other demand functions: logarithmic regression models

Logarithms convert changes in variables into percentage changes
(a) Linear demand function:

What are a, b, c and d?

What are
,
and
?
(b) Linear-log demand function:

A 1% change in px is associated with a change of x by 0.01b



If

A 1% change in py is associated with a change of x by 0.01c

A 1% change in I is associated with a change of x by 0.01d
, then
(c) Log-linear demand function:

A change in px by 1 unit is associated with a (100)b% change in x




A change in py by 1 unit is associated with a (100)c% change in x

A change in I by 1 unit is associated with a (100)d % change in I
(d) Log-log demand function:
(constant elasticity model)

A 1% change in px is associated with a b% change in x (price elasticity of demand)





= price elasticity of demand


A 1% change in py is associated with a c% change in x (cross price elasticity)
= cross price elasticity of demand


A 1% change in I is associated with a d% change in x (income elasticity)
= income elasticity of demand
4
Basic mathematical tools



Properties of linear functions:
Intercept = a
Slope
=b
 marginal effect of x on y is constant and is equal to b

Natural Logarithm
(when

Differential Calculus
Let

Partial derivative :
Let
is small)
, holding
constant
Example:
5