The Marshallian Hicksian and Slutsky Demand Curves Graphical

We start with the following diagram y In this part of the diagram we

Next we draw in the indifference curves showing the consumers tastes for x and

Recall the slope of the budget constraint is: y y 0 px x 0

From the initial equilibrium we can find the first point on the demand curve

Next consider a rise in the price of x, to px 1, . This

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

Our next exercise involves giving the consumer enough income so that they can reach

We derive the Hicksian Demand curve by projecting the demand for x downwards into

But now the consumer doesn’t have to consume x 0 y 0 y U

This diagram is going to get quite messy now and I apologise for that.

Summary px H S We can derive three demand curves on the basis of

px H S 1. The normal Marshallian Demand Curve M x

px H M S 2. The Hicksian compensated demand curve where agents are given

px H M S 3. The Slutsky income compensated demand curve where agents have

px H M S Finally, for a normal good the Marshallian demand curve is

Problems to think about • 1) Consider the shape of the curves if x

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The Marshallian, Hicksian and Slutsky Demand Curves Graphical Derivation

We start with the following diagram y 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 px x 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

Next we draw in the indifference curves showing the consumers tastes for x and y. y y 0 px x 0 x Then we draw in the budget constraint and find the initial equilibrium

Recall the slope of the budget constraint is: y y 0 px x 0 x

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

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 y y 0 px x 1 x 0 x 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. px 1 px 0 x 1 x 0 This shows us the new level of demand at p 1 x

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

Our next exercise involves giving the consumer enough income so that they can reach their original level of utility U 2 y y 0 U 2 px x 1 U 1 x x 0 px 1 px 0 Dx x 1 x 0 So we take the new budget constraint. . . And gradually increase the agents income, moving the budget constraint out. . . until we reach the indifference curve U 2

y y 0 px x 1 U 1 x x 0 px 1 px 0 Dx x 1 x 0 The new point of tangency tells us the demand for x when the consumer had been compensated so they can still achieve utility level U 2, but the relative price of x and y has risen to px 1/py 0.

y y 0 U 2 px U 1 x x 1 x. H x 0 px 1 px 0 Dx x 1 x 0 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 x. H

We derive the Hicksian Demand curve by projecting the demand for x downwards into the demand curve diagram y y 0 px U 2 U 1 x x 1 x. H x 0 px 1 px 0 Dx x 1 x. H x 0 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 x 0 px 0

y y 0 U 2 U 1 x px Hx px 1 px 0 We label the curve Hx Dx x 1 x. H x 0 Notice that the Hicksian Demand Curve is steeper than the Marshallian demand curve, when the good is a normal good

y y 0 U 2 px x x 0 px 1 px 0 Dx x 1 x. H x 0 Notice that an alternative compensation scheme would be to give the consumer enough income to buy their original bundle of goods, x 0 yo In this case the budget constraint has to moved out even further until it goes through the point x 0 y 0

But now the consumer doesn’t have to consume x 0 y 0 y U 3 y 0 U 2 px U 1 x x 0 px 1 px 0 Dx x 1 x. H x 0 So they will choose a new equilibrium point. . On a higher indifference curve

This diagram is going to get quite messy now and I apologise for that. I could knock. Once out the Hicksian again we findcurve the to make it clearer but Ifor want be demand x atyou thistonew able to see where lies of relative to by the higher itlevel income new one I am about to derive dropping a line down from U 3 the new equilibrium point to the x axis. y y 0 px Hx U 2 x xs x 0 px 1 px 0 Dx x 1 x. H x 0 xs We call this xs. It is the Slutsky demand. Once again this income compensated demand is measured at the price px 1

y U 3 y 0 px Mx Hx Sx U 2 x x 0 px 1 px 0 Dx x 1 x. H x 0 xs Finally, once again we can draw the Slutsky compensated demand curve through this new point xspx 1 and the original x 0 px 0 The new demand curve Sx is steeper than either the Marshallian or the Hicksian curve when the good is normal

Summary px H S We can derive three demand curves on the basis of our indifference curve analysis. M x

px H S 1. The normal Marshallian Demand Curve M x

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

px H M S 3. The Slutsky income compensated demand curve where agents have sufficient income to purchase their original bundle x

px H M S Finally, for a normal good the Marshallian demand curve is flatter than the Hicksian, which in turn is flatter than the Slutsky demand curve. x

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?