The Marshall, Hicks and Slutsky Demand Curves

Graphical Derivation

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

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

xpx

x

y

Recall the slope of the

budget constraint is:

dy

dx

p

px

y

xpx

x

y

x0

y0

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

xpx

x

y

px0

Next consider a rise in the price of x, to px

1. This causes the budget constraint to swing in as – px

1/py0

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

xpx

x

y

px0

x1

px1

x1

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

xpx

x

y

px0

px1

x1 x0

Dx

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

px0

x1

x1

px1

Dx

x

y

px

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 px

1/py0.U1

x0

y0

x0

px0

x1

x1

px1

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

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

px0

x1

x1

px1

Dx

x

y

px

x

U1

U2

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

px0

x1

x1

px1

Dx

x

y

px

x

U1

U2

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 x0y0Hx

xH

xH

x0

y0

x0

px0

x1

x1

px1

Dx

x

y

px

x

U1

U2

But now the consumer doesn’t have to consume

x0y0

xH

x0

y0

x0

px0

x1

x1

px1

Dx

x

y

px

x

U1

U2

U3

So they will choose a new equilibrium point on a higher indifference curve.

Hx

U3

xH

x0

y0

x0

px0

x1

x1

px1

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 px1

xs

xs

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

px0

x1

px1

Dx

x

y

px

x

U1

Hx

xs

xs

U2

Sx

M

HS

px

x

We can derive three demand curves on the

basis of our indifference curve analysis.

Summary

1. The normal Marshallian demand curve2. 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.

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?