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The Marshall, Hicks and Slutsky Demand Curves Graphical Derivation

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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 P x. This is effectively the space in which we draw the demand curve. We start with the following diagram: x y pxpx x

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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. x0x0 y0y0 x pxpx x y

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Recall the slope of the budget constraint is: x pxpx x y x0x0 y0y0

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From the initial equilibrium we can find the first point on the demand curve Projecting x 0 into the diagram below, we map the demand for x at p x 0 x0x0 y0y0 x pxpx x y px0px0

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Next consider a rise in the price of x, to p x 1. This causes the budget constraint to swing in as – p x 1 /p y 0 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 p 1 x x0x0 y0y0 x pxpx x y px0px0 x1x1 px1px1 x1x1

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We are now in a position to draw the ordinary demand curve. First we highlight the p x 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. y0y0 x pxpx x y px0px0 px1px1 x1x1 x0x0 DxDx

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Our next exercise involves giving the consumer enough income so that they can reach their original level of utility U 2. U2U2 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 U 2 U1U1 x0x0 y0y0 x0x0 px0px0 x1x1 x1x1 px1px1 DxDx x y pxpx 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 U 2, but the relative price of x and y has risen to p x 1 /p y 0. U1U1 x0x0 y0y0 x0x0 px0px0 x1x1 x1x1 px1px1 DxDx x y pxpx x U2U2 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. xHxH

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xHxH xHxH 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 p x 1. To get the Hicksian demand curve we connect the new point to the original demand x 0 p x 0 x0x0 y0y0 x0x0 px0px0 x1x1 x1x1 px1px1 DxDx x y pxpx x U1U1 U2U2

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Notice that the Hicksian demand curve is steeper than the Marshallian demand curve when the good is a normal good. We label the curve H x HxHx xHxH xHxH x0x0 y0y0 x0x0 px0px0 x1x1 x1x1 px1px1 DxDx x y pxpx x U1U1 U2U2

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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 x 0 y o In this case the budget constraint has to move out even further until it goes through the point x 0 y 0 HxHx xHxH xHxH x0x0 y0y0 x0x0 px0px0 x1x1 x1x1 px1px1 DxDx x y pxpx x U1U1 U2U2

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But now the consumer doesnt have to consume x 0 y 0 xHxH x0x0 y0y0 x0x0 px0px0 x1x1 x1x1 px1px1 DxDx x y pxpx x U1U1 U2U2 U3U3 So they will choose a new equilibrium point on a higher indifference curve. HxHx

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U3U3 xHxH x0x0 y0y0 x0x0 px0px0 x1x1 x1x1 px1px1 DxDx x y pxpx x U1U1 U2U2 HxHx 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 x s. It is the Slutsky demand. Once again this income compensated demand is measured at the price p x 1 xsxs xsxs

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Finally, once again we can draw the Slutsky compensated demand curve through this new point x s p x 1 and the original x 0 p x 0 The new demand curve S x is steeper than either the Marshallian or the Hicksian curve when the good is normal. U3U3 x0x0 y0y0 px0px0 x1x1 px1px1 DxDx x y pxpx x U1U1 HxHx xsxs xsxs U2U2 SxSx

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M H S pxpx x We can derive three demand curves on the basis of our indifference curve analysis. Summary 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. Finally, for a normal good the Marshallian demand curve is flatter than the Hicksian, which in turn is flatter than the Slutsky demand curve.

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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?

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