Download MICRO REVIEW: Slutsky (no, it doesn`t go away...)

The Slutsky Equation
or
“The Fundamental
Equation of the
Theory of Value”
Eugene Slutsky 1880 – 1948
Sulla teoria del bilancio del consumatore (1915)
Sir John R. Hicks, 1904-1989
Value and Capital (1939)
Why is this so important?
1. It describes the key predictions from our economic model.
2. It forms the foundation for modern “marginalist” economics.
3. Good practice with the fundamental skill of going from words
to pictures to math!
Read more:
http://homepage.newschool.edu/het//profiles/hicks.htm
and links from here…
What happens to the demand for a good when the price changes?
Would you buy more or fewer apples if the price goes up?
What if you grew apples?
What if your wage went from
$10 to $100
Would you work more or less?
Substitution effect
+
Income effect
=
Total (observed) effect!
The Slutsky Equation:
Total effect
= substitution effect
– income effect
Read more:
http://homepage.newschool.edu/het//profiles/slutsky.htm
xi xiU
xi

 xi
pi pi
M
Total effect: The observed change in demand due to a
change in price
=
Substitution effect: The change in demand due to a change
in the rate of exchange between two goods
holding purchasing power (utility) constant
Income effect: The change in demand due to a change in
purchasing power
xi x
xi
Why is this so important?

 xi
pi pi
M
U
i
The (own-price) substitution effect must be negative: If the price
of a good increases consumers will substitute to other goods and
the demand will go down.
The Law of Demand: If the demand for a good increases when
income increases (e.g. it is a normal good), then the demand for
that good must decrease when its price increases.
An Intuitive Derivation of the Slutsky Equation
From Louis Phlips Applied Consumption Analysis (1990) p. 40 - 42
If price changes by Δp, how much do you need to change income so you
could still buy the same amount?
M  xi pi
M
 xi
pi
The change in income is the quantity of the good times the change in price
Now there are two things changing: price and income:
xi
xi
xi 
pi 
M
pi
M
This is the response to a
compensated price
change!
xi xi xi M


pi pi M pi
xi xiU
xi

 xi
pi pi
M
“total derivative”
Divide through by Δp
A Numerical Illustration of Slutsky Substitution
From Hal Varian Intermediate Microeconomics (2003) p. 140 - 141
M
x1  10 
10 p1
Suppose your weekly demand for milk is:
If you have $120 per week and milk is $3 per gallon, how many gallons per
week do you buy?
120
x1  10 
 14
10  3
What is the change in quantity demanded if the price drops to $2 per gallon?
120
x1  10 
 16
10  2
2 is the total observed
change in quantity
demanded for a change
in price
16 14  2
xi xiU
xi

 xi
pi pi
M
A Numerical Illustration Continued…
How much less money do you need to
buy your original 14 gallons per week?
Old price: $3/gallon
M
x

10

1
New price: $2/gallon
10 p1
Old demand: 14
New demand: 16
Change in quantity demanded: 2
M  xi pi  14  2  3  14
What would be your demand at the new price if you reduce your income by
$14 per week?
120  14
106
x1  10 
 10 
 15.3
10  2
20
The substitution effect holding utility constant is:
Even when we take away income,
the relative price of milk is still
cheaper than other goods, so
demand goes up!
15.3 14  1.3
The Income effect is:
15.3 16  0.7
Total (observed) effect:
2  1.3   0.7 
Test yourself: do the example when price increases from $2 to $3
Slutsky Illustration with Pictures: the “pivot-shift” method
Y
original budget line
-$14
“compensated budget line”
“true new budget line”
14
15.3
16
milk
Slutsky Illustration with Pictures: the rotation method
(Hicks Substitution)
Y
For small changes in price the
Slutsky Substitution (pivot-shift)
is EQUAL to the
Hicksian Substitution (rotation).
original budget line
“compensated budget line”
ROTATES AROUND THE
ORIGINAL INDIFFERENCE
CURVE
“true new budget line”
14
15.3
16
milk
Slutsky Illustration for an Inferior Good
Demand decreases when income increases
Y
If the income effect outweighs the
substitution effect then we have a
Giffen Good:
demand falls when price declines!
This is very rare!!
original budget line
“compensated budget line”
“true new budget line”
X1
X2 XS
X
Test Yourself:
Draw the Slutsky substitution lines for goods that are perfect
complements. Is the total effect all substitution? All income? A
little of both?
Draw the Slutsky substitution lines for goods that are perfect
substitutes. Is the total effect all substitution? All income? A little
of both?
Another Numerical Illustration
Recall our example with coffee and bagels where:
MUc = 1/2B, MUb = 2C
Pc = $1, Pb = $2, M = $6
Optimal (2.4,1.2)
What happens if the price of coffee increases to $1.50?
What is the new optimal bundle?
What is the total effect of the price change?
C = 1/3B
1.5(1/3B) + 2B = 6
B = 2.4, C = .8
Old (2.4, 1.2) = new (2.4, .8) = (0, -.4)
How much more money would you need to buy the old bundle at
the new prices? $0.5 x 1.2 cups = $0.60
If you had this additional money what
would be your new optimal bundle (at the
new prices)?
C = 1/3B
1.5(1/3B) + 2B = 6.6
B = 2.64, C = .88
Another Numerical Illustration Continued…
Original bundle:
(2.4, 1.2)
New bundle:
(2.4, 0.8)
Compensated bundle: (2.64, 0.88)
What is the total effect, substitution effect and income effect of
the increase in coffee price?
Total = Original - New:
(2.4, 1.2) - (2.4, 0.8) = (0, -.4)
Substitution = Original – Compensated: (2.4, 1.2) - (2.64, 0.88) = (.24, -.32)
Income = Total – Substitution:
(0, -.4) – (.24, -.32) = (-.24, -.08)
Test yourself:
1. Draw the graphs for this problem.
2. Compute the effects and draw the graphs if coffee and bagels are
perfect complements at a ratio of 2 cups to 1 bagel.
3. Compute the effects and draw the graphs if coffee and bagels are
perfect substitutes at a ratio of 2 cups to 1 bagel.
Why do we care?
We can make predictions based on the theory!
Remember: all models are wrong but some are useful…
1.
2.
3.
4.
Own price substitution effects must be negative
Income effects for normal goods must be positive
Income effects for inferior goods must be negative
Cross-price effects must be symmetrical
George Box
These are called “general restrictions”
and have been critical for empirical microeconomics!
“Do we mean to say that observed demand behavior does in fact always
satisfy these conditions?...there is no reason why measured behavior should
obey them, as theory is always a simplification of reality…All we can hope is
that rough estimates, computed without imposing these constraints, will not
be inconsistent with them.” -- Phlips (1990) p. 53 - 54