Transcript The Demand Curves - Economics Network

The Marshall, Hicks 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.
x
px
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
y
Next we draw in the
indifference curves
showing the consumers’
tastes for x and y.
y0
x0
x
px
x
Then we draw
in the budget
constraint and
find the initial
equilibrium.
Recall the
slope of the
budget
constraint is:
y
y0
px
dy

dx
py
x0
x
px
x
From the initial equilibrium we
can find the first point on the
demand curve
y
y0
x
px
Projecting x0 into the
diagram below, we
map the demand for
x at px0
px0
x0
x
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
x1
x
To find the demand for
x at the new price we
locate the new
equilibrium quantity of x
demanded.
px
Then we drop a line
down from this point to
the lower diagram.
px1
px0
This shows us the new
level of demand at p1x
x1
x0
x
y
We are now in a position to draw
the ordinary demand curve.
y0
x
px
px1
This is the
Marshallian demand
curve for x.
Dx
px0
x1
x0
First we highlight the
px and x
combinations we
have found in the
lower diagram and
then connect them
with a line.
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
x1
px
x
x0
px1
px0
Dx
x1
x0
x
To do this we take
the new budget
constraint and
gradually increase
the agent’s income,
moving the budget
constraint out until
we reach the
indifference curve U2
y
y0
U2
U1
px
x1 xH
x
x0
px1
px0
Dx
x1
x0
x
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.
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.
We derive the Hicksian
demand curve by projecting
the demand for x
downwards into the
demand curve diagram.
y
y0
U2
U1
px
x1 xH
x
x0
px1
px0
To get the Hicksian
demand curve we
connect the new point to
the original demand x0px0
Dx
x1 xH x0
Notice this is the
compensated
demand for x when
the price is px1.
x
y
We label the curve Hx
y0
U2
U1
px
x1 xH
x
x0
Notice that the Hicksian
demand curve is
steeper than the
Marshallian demand
curve when the good is
a normal good.
px1
px0
Dx
Hx
x1 xH x0
x
y
y0
U2
U1
px
x1 xH
x
x0
In this case the
budget constraint
has to move out
even further until it
goes through the
point x0y0
px1
px0
Dx
Hx
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
x
y
But now the
consumer doesn’t
have to consume
x0y0
U3
y0
U2
U1
px
x1
x
x0
So they will choose
a new equilibrium
point on a higher
indifference curve.
px1
px0
Dx
Hx
x1 xH x0
x
Once again we find the demand for
x at this new higher level of income
by dropping a line down from the
new equilibrium point to the x axis.
y
U3
y0
U2
We call this xs . It is
the Slutsky demand.
U1
px
x
x1 xs x0
Once again this
income compensated
demand is measured
at the price px1
px1
px0
Dx
Hx
x1 xHxs x0
x
y
U3
y0
U2
Finally, once again
we can draw the
Slutsky compensated
demand curve
through this new
point xspx1 and the
original x0px0
U1
px
x
x1 xs x0
The new demand
curve Sx is steeper
than either the
Marshallian or the
Hicksian curve when
the good is normal.
px1
px0
Dx
xs
Hx
Sx
x
Summary
S
H
px
M
We2.
can
derive
three
1. The
normal
Marshallian
The
Hicksian
3. Thefor
Slutsky
income
Finally,
a normal
good
demand
curves
on
the
demand
curve
compensated
demand
compensated
demand
the
Marshallian
demand
basis
of
our
indifference
curve
where
agents
are
curve
where
agents
have
curve
is
flatter
than
the
curve
analysis.
given
sufficient
income
sufficient
income
to tois
Hicksian,
which
in turn
maintain
them
their
purchase
their
original
flatter than
theon
Slutsky
original
utilitycurve.
curve.
bundle.
demand
x
Problems to consider
1. Consider the shape of the curves if X is an inferior good.
2. Consider the shape of each of the curves if X is a Giffen
good.
3. Will it matter if Y is a Giffen or an inferior good?