Transcript Lecture 9: Marshall-Hicks

The Marshallian, Hicksian
and Slutsky Demand
Curves
Graphical Derivation
We start with the following diagram
y
In this part of the diagram we have drawn
the choice between x on the horizontal
axis and y on the vertical axis. Soon we
will draw an indifference curve in here
px
x
Down below we have drawn the
relationship between x and its price Px. This
is effectively the space in which we draw
the demand curve.
x
Next we draw in the
indifference curves
showing the consumers
tastes for x and y.
y
y0
px
x
x0
Then we draw
in the budget
constraint and
find the initial
equilibrium
Recall the slope of
the budget
constraint is:
y
y0
px
x
x0
px
dy

dx
py
From the initial equilibrium we
can find the first point on the
demand curve
y
y0
x
px
x0
Projecting x0 into the
diagram below, we
map the demand for
x at p0x
px0
x0
Next consider a rise in the price of
x, to px1,. This causes the budget
constraint to swing in as -px1/py0
is greater
y
y0
x
px
x1
x0
To find the demand for
x at the new price we
locate the new
equilibrium quantity of
x demanded.
Then we drop a line down
from this point to the
lower diagram.
px1
px0
x1
x0
This shows us the new level
of demand at p1x
We are now in a position to draw
the ordinary Demand Curve
y
y0
x
px
x1
x0
px1
First we highlight
the the px and x
combinations we
have found in the
lower diagram.
And then connect them
with a line.
px0
Dx
x1
x0
This is the Marshallian
demand curve for x
Our next exercise involves
giving the consumer enough
income so that they can reach
their original level of utility U2
y
y0
U2
U1
x
px
x1
x0
px1
px0
Dx
x1
x0
So we take the new
budget constraint...
And gradually increase
the agents income,
moving the budget
constraint out...
...until we reach the
indifference curve U2
y
y0
U1
x
px
x1
x0
px1
px0
Dx
x1
x0
The new point of
tangency tells us the
demand for x when
the consumer had
been compensated so
they can still achieve
utility level U2, but
the relative price of x
and y has risen to
px1/py0.
y
y0
U2
U1
x
px
x1 xH x0
px1
px0
Dx
x1
x0
The level of
demand for x
represents the
pure substitution
effect of the
increase in the
price of x
This is called the
Hicksian demand
for x and we will
label it xH
y
We derive the Hicksian
Demand curve by
projecting the demand for
x downwards into the
demand curve diagram
y0
U2
U1
x
px
x1 xH x0
px1
px
0
Dx
x1
xH x0
Notice this is the
compensated demand
for x when the price is
px1
To get the Hicksian demand
curve we connect the new
point to the original demand
x0px0
y
y0
U2
U1
x
px
Hx
We label the curve Hx
px1
px0
Dx
x1
xH x0
Notice that the Hicksian
Demand Curve is steeper than
the Marshallian demand
curve, when the good is a
normal good
y
y0
U2
x
px
x0
px1
px0
Dx
x1
xH x0
Notice that an alternative
compensation scheme would
be to give the consumer
enough income to buy their
original bundle of goods,
x0yo
In this case the budget
constraint has to moved out
even further until it goes
through the point x0y0
But now the
consumer
doesn’t have to
consume x0y0
y
U3
y0
U2
U1
x
px
x0
px1
px0
Dx
x1
xH x0
So they will
choose a new
equilibrium
point .. On a
higher
indifference
curve
This diagram is going to get quite
messy now and I apologise for that. I
could knockOnce
out the
Hicksian
again
we findcurve
the to
make it clearer
but Ifor
want
be
demand
x atyou
thistonew
able to see where
lies of
relative
to by
the
higher itlevel
income
new one I am
about to
derive
dropping
a line
down from
U3 the new equilibrium point to
the x axis.
U2
We call this xs . It is
the Slutsky demand.
x
y
y0
px
Hx
xs x0
px1
px0
Dx
x1
xH x0
xs
Once again this
income
compensated
demand is
measured at the
price px1
y
U3 Finally, once again we can
draw the Slutsky
compensated demand
U2
curve through this new
x point xspx1 and the original
x0px0
y0
px
Mx Hx Sx
x0
px1
px0
Dx
x1
xH x0
xs
The new demand curve Sx is
steeper than either the
Marshallian or the Hicksian
curve when the good is normal
Summary
S
px
We can derive three demand
curves on the basis of our
indifference curve analysis.
H
M
x
S
px
1. The normal Marshallian
Demand Curve
H
M
x
S
px
H
M
2. The Hicksian
compensated demand curve
where agents are given
sufficient income to
maintain them on their
original utility curve.
x
S
px
H
M
3. The Slutsky income
compensated demand
curve where agents
have sufficient income
to purchase their
original bundle
x
S
px
H
M
Finally, for a normal good
the Marshallian demand
curve is flatter than the
Hicksian, which in turn is
flatter than the Slutsky
demand curve.
x
Problems to think about
• 1) Consider the shape of the curves if x
is an inferior good.
• 2) Consider the shape of each of the curves
x is a Giffen good.
• 3) Will it matter if y is a Giffen or an
inferior good?