The Marshallian, Hicksian and Slutsky Demand

Curves

Graphical Derivation

x

x

y

px

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

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

y

px x0

y0

Recall the slope of the budget constraint is:

dy

dx

p

px

y

x

y

px 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 p0

x

x0

px0

x

y

px x0

y0

x0

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.

x1

x1

px1 Then we drop a line down

from this point to the lower diagram.

This shows us the new level of demand at p1

x

We are now in a position to draw the ordinary Demand Curve

x

y

px x0

y0

x0

px0

x1

x1

px1

First we highlight the the px and x combinations we have found in the lower diagram.

Dx

And then connect them with a line.

This is the Marshallian demand curve for x

x

y

px x0

y0

x0

px0

x1

x1

px1

Dx

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

U2So we take the new budget constraint...

And gradually increase the agents income, moving the budget constraint out...

...until we reach the indifference curve U2

U1

x

y

px x0

y0

x0

px0

x1

x1

px1

Dx

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

This is called the Hicksian demand for x and we will label it xH

x

y

px x0

y0

x0

px0

x1

x1

px1

Dx

The level of demand for x represents the pure substitution effect of the increase in the price of x

xH

U2

U1

x

y

px x0

y0

x0

px0

x1

x1

px1

Dx

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

U2U1

x

y

px

y0

x0

px0

x1

px1

Dx

xH

Notice that the Hicksian Demand Curve is steeper than the Marshallian demand curve, when the good is a normal good

We label the curve HxHx

U2

U1

Notice that an alternative compensation scheme would be to give the consumer enough income to buy their original bundle of goods,

x0yo

x

y

px

y0

x0

px0

x1

px1

Dx

xH

x0In this case the budget constraint has to moved out even further until it goes through the point x0y0

U2

x

y

px

y0

x0

px0

x1

px1

Dx

xH

x0

But now the consumer doesn’t have to consume x0y0

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

U3

U2

U1

x

y

px

y0

x0

px0

x1

px1

Dx

xH

x0

U3

U2

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.

xsHx

xs

Once again this income compensated demand is measured at the price px

1

This diagram is going to get quite messy now and I apologise for that. I could knock out the Hicksian curve to make it clearer but I want you to be able to see where it lies relative to the new one I am about to derive

x

y

px

y0

x0

px0

x1

px1

Dx

xH

x0

U3

U2

Hx

xs

Finally, once again we can draw the Slutsky compensated demand curve through this new point xspx

1 and the original x0px

0SxMx

The new demand curve Sx is steeper than either the Marshallian or the Hicksian curve when the good is normal

M

HS

px

x

We can derive three demand curves on the basis of our indifference curve analysis.

Summary

M

HS

px

x

1. The normal Marshallian Demand Curve

M

HS

px

x

2. The Hicksian compensated demand curve where agents are given sufficient income to maintain them on their original utility curve.

M

HS

px

x

3. The Slutsky income compensated demand curve where agents have sufficient income to purchase their original bundle

M

HS

px

x

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 think about

• 1) Consider the shape of the curves if x is an inferior good.

• 2) Consider the shape of each of the curves x is a Giffen good.

• 3) Will it matter if y is a Giffen or an inferior good?