This document outlines the graphical derivation of the Marshall, Hicks, and Slutsky demand curves. It begins by drawing indifference curves and a budget constraint on a graph with goods x and y. It then shows how shifting the budget constraint due to a price change in x generates points that define the Marshallian demand curve. The Hicksian curve is derived by compensating the consumer to maintain their original utility level. The Slutsky curve compensates the consumer to purchase their original bundle of goods. For normal goods, the Marshallian curve is flatter than the Hicksian, which is flatter than the Slutsky curve.

1. The Marshall, Hicks and
Slutsky Demand Curves
Graphical Derivation

2. 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.
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.
We start with the following diagram:
x
y
px
x

3. Next we draw in the
indifference curves
showing the consumers’
tastes for x and y.
Then we draw
in the budget
constraint and
find the initial
equilibrium.
x0
y0
x
px
x
y

4. Recall the
slope of the
budget
constraint is:
dy
dx
p
p
x
y
= −
x
px
x
y
x0
y0

5. From the initial equilibrium we
can find the first point on the
demand curve
Projecting x0 into the
diagram below, we
map the demand for
x at px
0
x0
y0
x
px
x
y
px
0

6. Next consider a rise in the price of
x, to px
1
. This causes the budget
constraint to swing in as – px
1
/py
0
is greater.
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.
This shows us the new
level of demand at p1
x
x0
y0
x
px
x
y
px
0
x1
px
1
x1

7. We are now in a position to draw
the ordinary demand curve.
First we highlight the
px and x
combinations we
have found in the
lower diagram and
then connect them
with a line.
This is the
Marshallian demand
curve for x.
y0
x
px
x
y
px
0
px
1
x1 x0
Dx

8. Our next exercise involves giving
the consumer enough income so
that they can reach their original
level of utility U2.
U2
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
U1
x0
y0
x0
px
0
x1
x1
px
1
Dx
x
y
px
x

9. 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 px
1
/py
0
.U1
x0
y0
x0
px
0
x1
x1
px
1
Dx
x
y
px
x
U2
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.
xH

10. xH
xH
We derive the Hicksian
demand curve by projecting
the demand for x
downwards into the
demand curve diagram.
Notice this is the
compensated
demand for x when
the price is px
1.
To get the Hicksian
demand curve we
connect the new point to
the original demand x0px
0
x0
y0
x0
px
0
x1
x1
px
1
Dx
x
y
px
x
U1
U2

11. Notice that the Hicksian
demand curve is
steeper than the
Marshallian demand
curve when the good is
a normal good.
We label the curve Hx
Hx
xH
xH
x0
y0
x0
px
0
x1
x1
px
1
Dx
x
y
px
x
U1
U2

12. 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 move out
even further until it
goes through the
point x0y0
Hx
xH
xH
x0
y0
x0
px
0
x1
x1
px
1
Dx
x
y
px
x
U1
U2

13. But now the
consumer doesn’t
have to consume
x0y0
xH
x0
y0
x0
px
0
x1
x1
px
1
Dx
x
y
px
x
U1
U2
U3
So they will choose
a new equilibrium
point on a higher
indifference curve.
Hx

14. U3
xH
x0
y0
x0
px
0
x1
x1
px
1
Dx
x
y
px
x
U1
U2
Hx
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.
We call this xs . It is
the Slutsky demand.
Once again this
income compensated
demand is measured
at the price px
1
xs
xs

15. Finally, once again
we can draw the
Slutsky compensated
demand curve
through this new
point xspx
1
and the
original x0px
0
The new demand
curve Sx is steeper
than either the
Marshallian or the
Hicksian curve when
the good is normal.
U3
x0
y0
px
0
x1
px
1
Dx
x
y
px
x
U1
Hx
xs
xs
U2
Sx

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

17. 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?