Micro 091021

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					        Micro 091021

Slutsky equation, Consumer surplus
                      Topics for this week

• Prove that the Hicksian demand function is the partial
  derivative of the Expenditure function
• Prove the Slutsky equation
• Introduce the elasticity notation for the Slutsky equation,
  and verifying the formula for the utility function discussed
  last week
• Discussing how the Slutsky equation holds for cross
  elasticities as well as for own elasticities
   – requires re-deriving the example discussed last week so that it
     includes the prices of both goods x and y (denoted by p and q)
• Introducing the idea of the symmetry of the Slutsky
  matrix, and how it imposes restrictions on demand
  functions
                   Topics for this week, cont’d

• Introducing the concept of consumer surplus (CS),
  starting with the diagrammatic and intuitive exposition
  (the Marshallian measure of consumer surplus)
• Ask the question: is the Marshallian CS an accurate
  measure of the consumer’s loss when the price of a
  good increases?
   – for a good with a substantial income effect, the answer is “no”
• Then ask the question: how can we get an accurate
  measure of the consumer’s loss of CS when a price
  increases
   – the answer turns out to be: using the indirect utility function, or
     the expenditure function
                Topics for this week, cont’d

• Then we associate the correct measure of the CS loss
  associated with a price increase with the area to the left
  of the Hicksian (not Marshallian), or compensated
  demand curve
• Next we consider the following question. Suppose the
  price of good x rises from p1 to p2, and that the
  consumer’s CS loss is A dollars. Now suppose the price
  of good x falls from p2 to p1. Does this mean that the
  consumer gains the same amount (A)?
• It turns out the answer is “no”. This leads to the concepts
  of the Equivalent Variation and the Compensating
  Variation
     More class notes and assignments for next week

• Your assignment for next week is as follows. Consider a
  consumer with the budget constraint px+qy=I, where p
  and q are the prices of goods x and y, respectively, and
  I is the consumer’s income. The consumer’s utility
  function is given by

             u  ln x  (1 / 3) ln y
• For this consumer, find the own-price elasticities and the
  cross-price elasticities of the demand for both goods,
  and verify the symmetry condition for this case
• Once you have done this exercise, I will post answers
  and further derivation notes
       Proving that the Hicksian demand function ..
• Consider x*,p*,u0, where x* is the solution to the
  expenditure minimization problem when the price vector
  is p*. Denote the expenditure function by E(p,u0)
• Now define G(p,u0)=E(p,u0)-px*. The function G is
  negative for all values of p that are different from p*
  (why?)
• The function reaches its maximum value (of G=0) at
  p=p*. But if it reaches a maximum value for that p-
  vector, that means that the derivatives of the G(p,u0)
  function with respect to each element of the p-vector
  are equal to zero there. This means
                G E
                     *  xi*  0
                pi* pi
               Hicksian demand function …

• But since x* is the vector that solves the expenditure
  minimization function, the last equation shows that the
  partial derivative of the expenditure function is the i’th
  element of that vector; that is, the demand function for
  the i’th good, as required
• The following three slides are from last week’s set
• Note that in this slide, we use h(j) to denote the Hicksian
  demand function for good j, and x(j) to denote the regular
  Marshallian demand function
                      Slutsky equation
• Let’s denote the regular (Marshallian) demand function
  for good i by
                     xi ( p, Y )
• Now the Slutsky equation states:

            xi hi        xi
                      xj     ,i, j
            p j p j      m

• Interpret the Slutsky equation: it divides the total effect of
  a price change into a substitution effect and an income
  effect
                 Proving the Slutsky equation
• The following equation is always true:
        h j ( p, u0 )  x j ( p, E ( p, u0 ))

• Now take the derivative of this with respect to pi and
  evaluate the derivative at u0, Y0

      h j ( p, u0 )       x j ( p, Y0 )       x j ( p, Y0 ) E ( p, u0 )
                                                                            
          pi                  pi                  Y                pi
      h j ( p, u0 )       x j ( p, Y0 )       x j ( p, Y0 )
                                                               xi
          pi                  pi                  Y
                      The Slutsky equation


• Rearranging the last expression gives the Slutsky
  equation. Note that it holds true as well when i and j are
  the same
• That is, it holds both when we evaluate the effect of a
  change in pi on xi (the own-price effect), but also when
  we evaluate the effect of a change in pi on xj, that is, a
  cross-price effect
• So the Slutsky equation just gives the subdivision of the
  effects of price changes into price and income effects,
  and holds both for own-price effects and cross-price
  effects
• In the case of 2 goods, it is based on the fact that along
  an indifference curve, utility is constant (at u0, say)
              Slutsky equation, elasticity version

• Remember that if z=f(x,y), the elasticity of z with respect to
  x and y can be written

                f ( x, y ) x         f ( x, y ) y
        zx                  , zy 
                   x       z            y       z
• After multiplying through by pi/xj we can therefore rewrite
  the Slutsky equation as


           x j pi              h j pi      pi x j I
                     x j pi           xi
           pi x j              pi x j      x j I I
             Slutsky equation, elasticity version

• But the right-hand side can be rewritten, if we use hj to
  denote the Hicksian demand function for the jth good,
  and si to denote the share of the consumer’s income
  that she spends on good i:
         x   j pi
                      h j pi  six j I

• Again, note that this holds for i=j, as well as when i
  and j are different.
• Next we show a restriction that must hold for cross-
  elasticities
             Slutsky equation, cross-elasticities

• Cross elasticities of Hicksian demand curves must
  satisfy the following restriction:
      s jh j ps  sshs p j ,s, j
• Note carefully what this says: the cross-elasticity of the
  Hicksian demand function for good j with respect to the
  price of good s, multiplied by the budget share of good j
  in total spending, must equal the cross-elasticity of the
  Hicksian demand function for good s with respect to the
  price of good j, multiplied by the budget share of good s
  in total spending
               Restriction on cross elasticities

• The reason for this is that the cross-price effects of the
  Hicksian demand functions must satisfy the following
  restrictions:

            h j hs
                    ,j, s
            ps p j
• This restriction, in turn, follows because h(j) is the partial
  derivative of the expenditure function with respect to the
  price of j. Thus the derivative on the left in the equation
  above can also be written

    h j   2 E ( p, u )  2 E ( p, u ) hs
                                     
    ps     p j ps       ps p j      p j
              Restriction on cross elasticities

• The reason why the cross-price effects are the same,
  therefore, is that both of them are second cross-partial
  derivatives of the expenditure function. But for
  continuous differentiable functions, the order in which
  you perform the differentiation when finding cross-partial
  derivatives doesn’t matter, so the two cross-partial
  derivatives in the middle of the above formulae are the
  same. Therefore, the cross-price effects are the same
• The restriction on the cross-elasticities can then be
  derived by using the definition of elasticities
                     Consumer surplus

• The concept of consumer surplus arises because with
  convex preferences, consumers in competitive markets
  are able to buy inframarginal units of individual goods at
  a price that is lower than she would be willing to pay for
  them
• Definition of “inframarginal” units: all units that the
  consumer would hypothetically buy even if the price of
  the good was higher than it actually is
• In equilibrium, the consumer buys enough units so that
  her willingness to pay for the last (marginal) unit is equal
  to the price.
                 Consumer surplus, cont’d

• Since (with convex preferences) her willingness to pay
  declines as she buys more units, the willingness to pay
  for earlier (inframarginal) units must therefore be higher
  than the market price
• The diagram on the next page illustrates an individual
  consumer’s demand curve for a good, assuming the
  consumer has a given income and the prices of other
  goods remain constant
• If the market price of the good is p0, the consumer buys
  the quantity indicated at Q. If the price were higher, the
  consumer would buy smaller quantities, but she would
  still buy at least some units as long as the price were
  less than p*
                   Consumer surplus, cont’d

• The conventional way to measure consumer surplus now
  is as area A in the diagram on the following slide. Note
  several things about this area
• First, it can also be interpreted as the sum of the
  surpluses that the consumer enjoys on discrete units of
  the good (see the textbook). If each unit is very small,
  this sum is approximately the same as the area under
  the curve. It can then be evaluated as an integral
   – Note that there are two ways of evaluating the integral
• Note also that the area that measures the consumer
  surplus is measured in money (dollars), not “utility”
P        Consumer Surplus (CS) is then
         given by area A in the figure,
         which in turn can be written as:
P*


         CS  0.5Q( P *  P 0 )
     A




P0



                                            D


                            Q
                Consumer surplus, cont’d

• Note that the consumer surplus is most interesting when
  we consider something that changes it
• Specifically, suppose the price of the good in the
  previous diagram rises from P0 to P1. Then the
  consumer surplus changes to area A’<A. The difference
  is area B+C. It can be measured in dollars, and can be
  interpreted as the consumer’s loss from the price
  increase
• But on the earlier slides, we asked the question: is this
  area an accurate measure of the consumer’s loss?
• In general, the answer is “no”. Let’s try to see why
               When price rises from P0 to P1,
               Consumer Surplus (CS), changes to area A’,
P
               which is smaller than A. It is given by:


 P*
                CS  0.5Q1 ( P*  P1 )

                                  The change in CS is equal to the
      A’                          difference, area (B+C)


P1

           B                  C
P0




                                                             D


                         Q1               Q0
                 Consumer surplus, cont’d

• To see this, note that if the price of the good rises from
  p0 to p1, the consumer’s utility is reduced
• An accurate measure of the consumer’s loss could then
  be based on an answer to the following question: if the
  price of this good were to rise, how much money would it
  take to compensate the consumer for this loss?
• One way to measure the consumer surplus loss is on the
  basis of the indirect utility function V(p,I)
• Specifically, we can ask the following question: by how
  much would we have to increase the consumer’s income
  when the price of this good rises from P0 to P1, in order
  to leave the consumer at the same level of utility as
  before?
                Consumer surplus, cont’d

• But when we define it this way, we can in fact compute
  the loss of consumer surplus directly from the
  expenditure function
• But here comes an interesting point: if we interpret the
  consumer surplus measure this way, it will in fact be
  measured correctly by area B+C only if the demand
  curve D in the diagram is the Hicksian (compensated)
  demand curve
• To see this, consider the CS loss from a very small price
  increase from P0; call it dP
• The loss from a small price increase dP is just the dP
  times the derivative of the expenditure function
• But the derivative of the expenditure function is the
  compensated demand function!
  Consumer surplus and compensated demand functions

• So the number we get if we take the (horizontal) integral
  of the area between the old and the new price, to the left
  of the compensated (Hicksian) demand curve, is in fact
  the change in the expenditure function that is necessary
  to compensate the consumer for a price increase from
  p0 to p1
• But here comes one further question: suppose we ask
  the question: if we start at p1, and then reduce the price
  of the good to p0, what is an accurate measure of the
  consumer’s gain from this price decrease?
• At first glance, it would seem that the answer should be:
  “the same as the loss from a price increase from p0 to
  p1”
                Consumer surplus, cont’d

• Unfortunately, this answer turns out to be wrong. In the
  next little while, we will discuss why it is wrong
• (Note one thing first: If we measure consumer surplus
  using the area to the left of the Marshallian demand
  curve between the old and the new price, the two are the
  same. Thus, the fact that the loss from a price increase,
  when correctly measured, is not of the same size as the
  gain from the same change in the other direction, must
  have something to do with the fact that we are
  measuring the gains and losses using Hicksian demand
  curves)
• First, we do an example
         Hicksian and Marshallian demand curves

• From the example, we can see that in this case, the
  consumer’s loss from a price increase is larger than the
  consumer’s gain from a price decrease
• The reason in this case is as follows. There is a different
  Hicksian demand curve through each point on the
  Marshallian demand curve
   – The different Hicksian demand curve don’t cross. Why?
• Since the CS measure is the area to the left of the
  Hicksian demand curve between the old and the new
  price, when the demand curve shifts the CS measure
  changes
                Consumer surplus, concluded

• In the example, the Hicksian demand curve is steeper
  than the Marshallian demand curve. Is that always true
  (for other utility functions as well)?
• Ans: No. If the good that we are illustrating is an inferior
  good, the Hicksian demand curve is less steep than the
  Marshallian demand curve
• Finally, some people measure the consumer’s loss from
  a price increase by the area to the left of the Hicksian
  demand curve that goes through the point on the
  Marshallian demand curve at the higher price
   – We have called that measure “the consumer’s gain from a price
     decrease”
                Consumer surplus, concluded

• This alternative measure of the consumer’s loss from a
  price increase is referred to as the “Equivalent Variation”,
  whereas our measure of the loss is called the
  “Compensating Variation”
   – The equivalent variation is the answer to the question: at the
     initial price, how much would the consumer be willing to pay for a
     promise that the price would not increase?
   – The compensating variation, in turn, is the answer to the
     question: how much money would we have to give the consumer
     after the price has increased in order to keep her at the same
     utility as at the lower price?
   – So the difference between the two measures is that we hold the
     consumer’s utility constant at two different levels

				
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