The Marshall, Hicks and Slutsky Demand Curves Graphical Derivation

We start with the following diagram: 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.

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

Recall the slope of the budget constraint is: x px y y0 x0

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

Then we drop a line down from this point to the lower diagram. 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 x1 x 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 x1 x0 x

This is the Marshallian demand curve for x. 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 x1 x0 x

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 agent’s income, moving the budget constraint out until we reach the indifference curve U2 y0 U2 U1 x x1 x0 px px1 px0 Dx x1 x0 x

This is called the Hicksian demand for x and we will label it xH. 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 xH x x1 x0 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 x1 x0

Notice this is the compensated demand for x when the price is px1. 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 xH x x1 x0 px px1 To get the Hicksian demand curve we connect the new point to the original demand x0px0 px0 Dx x x1 xH x0

y0 U2 U1 xH x1 x0 px1 px0 Dx Hx x1 xH x0 We label the curve Hx 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 x1 xH x0

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 xH x x1 x0 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 x1 xH x0

But now the consumer doesn’t have to consume x0y0 px So they will choose a new equilibrium point on a higher indifference curve. px1 px0 Dx Hx x x1 xH x0

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

U3 y0 U2 U1 x1 x0 xs px1 px0 Dx Hx Sx xs Finally, once again we can draw the Slutsky compensated demand curve through this new point xspx1 and the original x0px0 y0 U3 U2 U1 x x1 xs x0 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

1. The normal Marshallian demand curve 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. H 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. px M x

Problems to consider Consider the shape of the curves if X is an inferior good. Consider the shape of each of the curves if X is a Giffen good. Will it matter if Y is a Giffen or an inferior good?