Slutsky Equation
Lectures in Microeconomics-Charles W. Upton
The Slutsky Equation
These effects are
often summarized
in the Slutsky
equation
Q  Q 
 Q 

 Q


P  P U U o
 I 
SLutsky Equation
The Slutsky Equation
Q  Q 
 Q 

 Q


P  P U U o
 I 
SLutsky Equation
The Slutsky Equation
Q  Q 
 Q 

 Q


P  P U U o
 I 
The substitution effect is the change in
demand from a movement along the
indifference
curve.
SLutsky Equation
The Slutsky Equation
Q  Q 
 Q 

 Q


P  P U U o
 I 
The income effect is the change in
demand from the effective increase in
income
SLutsky Equation
A Caution
• The version of the Slutsky equation we use
is only an approximation.
SLutsky Equation
A Caution
• The version of the Slutsky equation we use
is only an approximation.
• We are assuming discrete changes in price
and income; the correct equation assumes
infinitesimal changes.
SLutsky Equation
Why spend time on this topic?
• Giffin Goods
SLutsky Equation
Why spend time on this topic?
• Giffin Goods
• The Demand for Leisure
SLutsky Equation
Why spend time on this topic?
• Giffin Goods
• The Demand for Leisure
– As wage rates increase, the cost of an hour of
leisure increases
– Demand goes up because the income effect
dominates the substitution effect.
SLutsky Equation
Why spend time on this topic?
• Giffin Goods
• The Demand for Leisure
• Different Slopes
.
SLutsky Equation
Why spend time on this topic?
• Giffin Goods
• The Demand for Leisure
• Different Slopes
– Changes in the price of one brand versus
changes in the prices of all brands.
.
SLutsky Equation
Why spend time on this topic?
• Giffin Goods
• The Demand for Leisure
• Different Slopes
– Changes in the price of one brand versus
changes in the prices of all brands.
– Heavily purchased goods versus lightly
purchased goods.
SLutsky Equation
Restating The Slutsky Equation
Q  Q 
 Q 

 Q


P  P U U o
 I 
SLutsky Equation
The Marshallian Demand Curve
Q  Q 
 Q 

 Q


P  P U U o
 I 
SLutsky Equation
The Marshallian Demand Curve
Q  Q 
 Q 

 Q


P  P U U o
 I 
SLutsky Equation
The Hicksian Demand Curve
Q  Q 
 Q 

 Q


P  P U U o
 I 
SLutsky Equation
The Hicksian Demand Curve
Q  Q 
 Q 

 Q


P  P U U o
 I 
SLutsky Equation
Sir John Hicks
SLutsky Equation
Sir John Hicks
The Hicksian
Demand Curve is the
right one to use for
consumer surplus
calculations, but we
generally use the
Marshallian one
SLutsky Equation
Sir John Hicks
The Hicksian
Why? The
Demand Curve is the
difference is
right one to use for
usually small
consumer surplus
calculations, but we
generally use the
Marshallian one
SLutsky Equation
A Demonstration
Q  Q 
 Q 

 Q


P  P U U o
 I 
Q
 Q 
 Q 

 Q



 P U U o P
 I 
SLutsky Equation
Multiplying Through
Q
 Q 
 Q 

 Q



 P U U o P
 I 
P Q P
 Q 
 Q  P

 Q



 P U U o Q P Q
 I  Q
I
SLutsky Equation
The Two Elasticities
Q
 Q 
 Q 

 Q



 P U U o P
 I 
P Q P
 Q 
 Q  P

 Q



 P U U o Q P Q
 I  Q
I
SLutsky Equation
The Elasticity Relationship
P Q P
 Q 
 Q  P

 Q



 P U U o Q P Q
 I  Q
 Q  P
 H   M  Q

 I  Q
P
P
SLutsky Equation
More Manipulation
P Q P
 Q 
 Q  P

 Q



 P U U o Q P Q
 I  Q
QP  Q  I
H M 


I  I  Q
p
p
SLutsky Equation
The Missing Terms
P Q P
 Q 
 Q  P

 Q



 P U U o Q P Q
 I  Q
QP  Q  I
  


I  I  Q
P
H
P
M
    
P
H
P
M
SLutsky Equation
I
The Final Relation
    
P
H
P
M
SLutsky Equation
I
The Final Relation
    
P
H
P
M
Unless ω is pretty
large, the difference is
small
SLutsky Equation
I
The
Housing
Final Relation
Leisure
    
P
H
P
M
Unless ω is pretty
large, the difference is
small
SLutsky Equation
I
End
©2006 Charles
W. Upton
SLutsky Equation