The Marshall, Hicks and Slutsky Demand Curves

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The Marshall, Hicks and Slutsky Demand Curves

• Graphical Derivation

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We start with the following diagram
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.
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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.
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Recall the slope of the budget constraint is
y0
x0
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From the initial equilibrium we can find the
first point on the demand curve
y0
Projecting x0 into the diagram below, we map the
demand for x at px0
px0
x0
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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
To find the demand for x at the new price we
locate the new equilibrium quantity of x
demanded.
y0
x
x1
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
x
x1
x0
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y
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.
y0
x
px
px1
This is the Marshallian demand curve for x.
Dx
px0
x
x1
x0
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Our next exercise involves giving the consumer
enough income so that they can reach their
original level of utility U2.
y
To do this we take the new budget constraint and
gradually increase the agents income, moving the
budget constraint out until we reach the
indifference curve U2
y0
U2
U1
x
x0
x1
px
px1
px0
Dx
x0
x1
x
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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
xH
x0
x1
The level of demand for x represents the pure
substitution effect of the increase in the price
of x.
px
px1
px0
Dx
This is called the Hicksian demand for x and we
will label it xH.
x
x0
x1
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We derive the Hicksian demand curve by projecting
the demand for x downwards into the demand curve
diagram.
y
y0
U2
Notice this is the compensated demand for x when
the price is px1.
U1
x
xH
x0
x1
px
px1
To get the Hicksian demand curve we connect the
new point to the original demand x0px0
px0
Dx
x
xH
x0
x1
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y
We label the curve Hx
y0
U2
U1
x
xH
x0
x1
px
Notice that the Hicksian demand curve is steeper
than the Marshallian demand curve when the good
is a normal good.
px1
px0
Dx
Hx
x
xH
x0
x1
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Notice that an alternative compensation scheme
would be to give the consumer enough income to
buy their original bundle of goods x0yo
y
y0
U2
U1
x
xH
x0
x1
px
In this case the budget constraint has to move
out even further until it goes through the point
x0y0
px1
px0
Dx
Hx
x
xH
x0
x1
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y
But now the consumer doesnt have to consume x0y0
y0
U2
U1
x
x0
x1
px
So they will choose a new equilibrium point on a
higher indifference curve.
px1
px0
Dx
Hx
x
xH
x0
x1
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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
We call this xs . It is the Slutsky demand.
U2
U1
x
x0
x1
xs
px
Once again this income compensated demand is
measured at the price px1
px1
px0
Dx
Hx
xs
x
xH
x0
x1
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Finally, once again we can draw the Slutsky
compensated demand curve through this new point
xspx1 and the original x0px0
y
U3
y0
U2
U1
x
x0
x1
xs
px
The new demand curve Sx is steeper than either
the Marshallian or the Hicksian curve when the
good is normal.
px1
px0
Dx
Hx
Sx
xs
x
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Summary
S
We can derive three demand curves on the basis of
our indifference curve analysis.
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.
H
Finally, for a normal good the Marshallian demand
curve is flatter than the Hicksian, which in turn
is flatter than the Slutsky demand curve.
px
M
x
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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?