The Marshall_ Hicks and Slutsky Demand Curves - The Economics

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The Marshall_ Hicks and Slutsky Demand Curves - The Economics Powered By Docstoc
					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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posted:3/3/2013
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