Advanced Microeconomics (3)

1
Advanced Microeconomics(3)

• Zheng Changde
• Southwest University for Nationalities

2
1 The Dual Problem

• 1.1 Expenditure Minimization
• 1. The expenditure minimization problem
• The solution to this problem tells us the minimum
  expenditure that is necessary such that the
  consumers utility is not lower than u.
• 2. Solving this problem tells the government the
  welfare payment necessary to keep a persons
  utility above a minimum level.

3

• 3. Definitions A solution to the expenditure
  minimization problem is known as the Hicksian
  demand function and is denoted by xh (p, u) .
• Substituting xh (p, u) into the objective
  function gives us the expenditure function e (p,
  u) p xh (p, u) .

4

• 4. Properties of Hicksian Demand Suppose u is
  continuous and strictly increasing, then xh has
  the following properties
• These properties are similar to those of the
  Marshallian demand. Briefly, 1 is obvious from
  the defn. of the minimization problem 2 follows
  from strict monotonicity.

5
(No Transcript)
6
(No Transcript)
7
(No Transcript)
8
(No Transcript)
9

• 9. Proof
• (a) The consumer can achieve u (0) by not
  spending anything. Since the minimum utility is
  be definition greater than u (0), the minimum
  expenditure that is necessary to achieve a
  utility greater than the lowest level is not
  greater than zero.Since prices are non-negative,
  the total expenditure is non-negative. Thus, the
  minimum expenditure is zero.
• (b) Intuitively, the result follows from the
  ontinuity of u and the continuity of the budget
  set.

10
(No Transcript)
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
1.2 Relationship Between Marshallian and Hicksian
Demand Functions

• 1. The indirect utility function and expenditure
  function are intimately connected. So are the
  Marshallian and Hicksian demand functions. We
  have already seen that the two demand functions
  share a common tangency condition. Regardless of
  whether ones objective is to maximize utility or
  minimize expenditure, one should consume in such
  a way that the marginal utility per dollar spent
  on each good is the same.

16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
(No Transcript)
20

• 9. There are two ways to derive the Hicksian
  demand. The direct way is to solve expenditure
  minimization problem. But we can also get the
  marshallian demand by solving the utility
  maximization problem. Substitute the demand back
  into the utility function. This gives the
  indirect utility function. From the indirect
  utility function, we know how much income need to
  change to keep utility constant as price change.
  Substitute this back into the Marshallian demand
  gives the Hicksian demand.

21
(No Transcript)
22
1.2.1 Income and Substitution Effects
23

• 2. To separate the income effect from the
  substitution effect, we can imagine that the
  consumers are compensated for the change in
  purchasing power due to a change in price so that
  their utility remain unchanged, and study how
  their demand for a good will change purely as a
  result of relative price changes. Hence, the
  Hicksian demand is also called the compensated
  demand, and the Marshallian Demand uncompensated
  demand

24
(No Transcript)
25
1.2.2 Giffen Goods
26

• 2. It is interesting that the idea of Giffen
  goods was first raised by Marshall in the third
  editions of Principles of Economics (1895)
• There are however some exceptions. For instance,
  as Mr. Giffen has pointed out, a rise in the
  price of bread makes so large a drain on the
  resources of the poorer labouring families and
  raises so much the marginal utility of money to
  them, that they are forced to curtail their
  consumption of meat and the more expensive
  farinaceous foods and, bread being still the
  cheapest food which they can get and will take,
  they consume more and not less of it. But such
  cases are rare when they are met with they must
  be treated separately.

27

• 3. Note that Marshall got exactly the right
  intuition with the help of the Slutsky equation
  which was developed later. Examples of Giffen
  goods are hard to come by. Economists used to
  cite potatoes during the Irish Potatoes
  (1845-1849) famine caused by a fungus that
  destroyed a large part the potato crop. Modern
  research has casted doubt on the claim. One
  reason is that since the total output of potato
  dropped significantly, it would be impossible for
  the poor to consume more potato as a group.
• 4. Aside The Mr. Giffen Marshall referred to is
  the economist Sir Robert Giffen. But there is no
  written record that Giffen had ever written
  anything about the goods that now bear his name.
• 5. The only know example of Giffen goods comes
  from experiments with rats.

28
1.3 The Slutsky Substitution Matrix
29
(No Transcript)
30
(No Transcript)
31

• 5. Theorem 2.5 (Reny pp.81) Symmetry and budget
  balancedness implies x is homogeneous of degree
  zero in p and y.
• 6. Proof Reny pp.81.

32
(No Transcript)
33
1.4 Compensating Variation and Consumer Surplus

• 1. When the government decides whether to cut or
  raise sales taxes, it has to know how will the
  tax affect consumer welfare. In antitrust law
  suit, an important factor that the court will
  look into is whether consumers are hurt by the
  monopolist.
• 2. The expenditure function provides us a formal
  way to measure how a change in price affects
  consumer welfare. Definition Compensating
  variation (CV) associated with a change in the
  price of good i is the income compensation
  required to keep the utility of a consumer
  constant. That is,

34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37

• 6. Two useful applications of this fact
• (a) Lump-sum tax is better than excise tax (tax
  per unit of output). An excise tax is equivalent
  to raising the price of good x. Under the new
  tax, the consumer will choose bundle x. Now
  suppose we replace the excise tax by a lump-sum
  tax that generates the same tax revenue. This
  means that x will still be feasible under the new
  budget constraint. Note that
• (px t)x pyy I
• ? pxx pyy I - tx.
• Graphically, this means that the new budget line
  will have the same slope as the original one and
  pass through x. It is clear from the diagram that
  the consumer must be better off. Excise tax is
  worse than a lump sum tax that generates the same
  tax revenue because it also distort the relative
  price of the goods.

38
(No Transcript)
39
(No Transcript)