# The Marshall_ Hicks and Slutsky Demand Curves - The Economics by zhangsshaohui123

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```									The Marshall, Hicks and
Slutsky Demand Curves

Graphical Derivation
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
Then we draw
in the budget
x0   x
constraint and
px                 find the initial
equilibrium.

x
Recall the
y
slope of the
budget
constraint is:

y0
dy    px

dx    py
x0   x
px

x
y          From the initial equilibrium we
can find the first point on the
demand curve

y0

x
px
Projecting x0 into the
diagram below, we
map the demand for
px0                              x at px0

x0      x
Next consider a rise in the price of
y                x, to px1. This causes the budget
constraint to swing in as – px1/py0
is greater.

y0                           To find the demand for
x at the new price we
locate the new
equilibrium quantity of x
x1              x                 demanded.
px
Then we drop a line
px1                          down from this point to
the lower diagram.
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.

First we highlight the
px and x
y0
combinations we
have found in the
lower diagram and
x       then connect them
px                                     with a line.

px1                                  This is the
Marshallian demand
px0             Dx                   curve for x.

x1   x0             x
Our next exercise involves
y                           giving the consumer enough
income so that they can reach
their original level of utility U2.

To do this we take
y0                                          the new budget
U2       constraint and
the agent’s income,
x          moving the budget
x1    x0
px                                        constraint out until
px1                                           we reach the
indifference curve U2
px0                   Dx

x1        x0              x
y                                              The new point of
tangency tells us the
demand for x when
been compensated so
they can still achieve
y0                                      utility level U2, but the
U2
relative price of x and
U1             y has risen to px1/py0.
x
x1 xH    x0                   The level of demand for
px
x represents the pure
px1                                  substitution effect of the
increase in the price of x.
px0                  Dx
This is called the
Hicksian demand for x
and we will label it xH.
x1       x0              x
We derive the Hicksian
y                            demand curve by projecting
the demand for x
downwards into the
demand curve diagram.

y0
U2     Notice this is the
compensated
U1
demand for x when
x
x1 xH   x0                         the price is px1.
px

px1                                   To get the Hicksian
demand curve we
px0                 Dx              connect the new point to
the original demand x0px0

x1 xH x0                x
y

We label the curve Hx

y0
U2
U1
x
x1 xH   x0
px                                    Notice that the Hicksian
demand curve is
px1                                       steeper than the
Marshallian demand
px0                 Dx                 curve when the good is
a normal good.
Hx
x1 xH x0                x
Notice that an
y
alternative
compensation
scheme would be
to give the
consumer enough
U2    original bundle of
U1                 goods x0yo
x
x1 xH   x0
px
In this case the
px1                                     budget constraint
has to move out
px0                 Dx                 even further until it
goes through the
point x0y0
Hx
x1 xH x0                x
y                                      But now the
consumer doesn’t
have to consume
x0y0

y0                     U3
U2
U1
x
x1    x0
px
So they will choose
px1                                  a new equilibrium
point on a higher
px0               Dx                 indifference curve.

Hx
x1 xH x0              x
y                      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.

y0                           U3
U2    We call this xs . It is
the Slutsky demand.
U1
x
x1 xs x0
px                                            Once again this
px1                                         income compensated
demand is measured
px0                                            at the price px1
Dx

Hx
x1 xHxs x0                   x
y                                    Finally, once again
we can draw the
Slutsky compensated
demand curve
through this new
y0                     U3             point xspx1 and the
U2       original x0px0

U1
x
x1 xs x0
px                                    The new demand
curve Sx is steeper
px1
than either the
Marshallian or the
px0               Dx                 Hicksian curve when
the good is normal.
Hx
Sx
xs                  x
Summary

S       We2. The Hicksian
derive
1. The canSlutsky three
normal Marshallian
3. The            income
H         Finally, for a normal good
demand curvesdemand  on the
px                        demand curve
compensated demand
compensated demand
the of our indifference
basis Marshallian
curve where agents are
M             curve where agents have
curve is analysis. the
curve flatter than to
given sufficient income is
sufficient income to
Hicksian, which in turn
maintain them on their
purchase their original
flatter than the Slutsky
original utility curve.
bundle.
demand curve.

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?

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