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The Marshall, Hicks 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. x px 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 y Next we draw in the indifference curves showing the consumers’ tastes for x and y. y0 Then we draw in the budget x0 x constraint and px find the initial equilibrium. x Recall the y slope of the budget constraint is: y0 dy px dx py x0 x px x y From the initial equilibrium we can find the first point on the demand curve y0 x px Projecting x0 into the diagram below, we map the demand for px0 x at px0 x0 x Next consider a rise in the price of y x, to px1. This causes the budget constraint to swing in as – px1/py0 is greater. y0 To find the demand for x at the new price we locate the new equilibrium quantity of x x1 x demanded. px Then we drop a line px1 down from this point to the lower diagram. px0 This shows us the new level of demand at p1x x1 x0 x y We are now in a position to draw the ordinary demand curve. First we highlight the px and x y0 combinations we have found in the lower diagram and x then connect them px with a line. px1 This is the Marshallian demand px0 Dx curve for x. x1 x0 x Our next exercise involves y giving the consumer enough income so that they can reach their original level of utility U2. To do this we take y0 the new budget U2 constraint and U1 gradually increase the agent’s income, x moving the budget x1 x0 px constraint out until px1 we reach the indifference curve U2 px0 Dx x1 x0 x y The new point of tangency tells us the demand for x when the consumer had been compensated so they can still achieve y0 utility level U2, but the U2 relative price of x and U1 y has risen to px1/py0. x x1 xH x0 The level of demand for px x represents the pure px1 substitution effect of the increase in the price of x. px0 Dx This is called the Hicksian demand for x and we will label it xH. x1 x0 x We derive the Hicksian y demand curve by projecting the demand for x downwards into the demand curve diagram. y0 U2 Notice this is the compensated U1 demand for x when x x1 xH x0 the price is px1. px px1 To get the Hicksian demand curve we px0 Dx connect the new point to the original demand x0px0 x1 xH x0 x y We label the curve Hx y0 U2 U1 x x1 xH x0 px Notice that the Hicksian demand curve is px1 steeper than the Marshallian demand px0 Dx curve when the good is a normal good. Hx x1 xH x0 x Notice that an y alternative compensation scheme would be to give the consumer enough y0 income to buy their U2 original bundle of U1 goods x0yo x x1 xH x0 px In this case the px1 budget constraint has to move out px0 Dx even further until it goes through the point x0y0 Hx x1 xH x0 x y But now the consumer doesn’t have to consume x0y0 y0 U3 U2 U1 x x1 x0 px So they will choose px1 a new equilibrium point on a higher px0 Dx indifference curve. Hx x1 xH x0 x y 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. y0 U3 U2 We call this xs . It is the Slutsky demand. U1 x x1 xs x0 px Once again this px1 income compensated demand is measured px0 at the price px1 Dx Hx x1 xHxs x0 x y Finally, once again we can draw the Slutsky compensated demand curve through this new y0 U3 point xspx1 and the U2 original x0px0 U1 x x1 xs x0 px The new demand curve Sx is steeper px1 than either the Marshallian or the px0 Dx Hicksian curve when the good is normal. Hx Sx xs x Summary S We2. The Hicksian derive 1. The canSlutsky three normal Marshallian 3. The income H Finally, for a normal good demand curvesdemand on the px demand curve compensated demand compensated demand the of our indifference basis Marshallian curve where agents are M curve where agents have curve is analysis. the curve flatter than to given sufficient income is sufficient income to Hicksian, which in turn maintain them on their purchase their original flatter than the Slutsky original utility curve. bundle. demand curve. x 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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