Part IIA, Paper 1 Consumer and Producer Theory

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Part IIA, Paper 1Consumer and Producer Theory

• Lecture 4
• Revealed Preferences and Consumer Welfare
• Flavio Toxvaerd

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Todays Outline

• Leftovers from Lecture 2
• Revealed preferences
• WARP, SARP, GARP
• Indices

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Primal Approach
Dual Approach
Duality
Integrability problem
Solve
Solve
Marshallian Demand x1(p,m) and x2(p,m)
Hicksian Demand
Equivalent if
Substitute into cost equation
Substitute into u(x,y)
Roys Identity
Shephards Lemma
Indirect Utility v(p,m)
Invert
Expenditure Function
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Consumer Surplus and Welfare
Following a price change,
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Consumer Surplus and Welfare
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Question

• Question What can we say about consumer
  preferences from observations of choice
  decisions?
• Recall rationality assumption Consumers select
  most preferred option available
• So observed selection has been revealed as
  preferred to all other available options

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Revealed Preference
If, at prices p1 , p2 and income m, a consumer
purchases a consumption bundle a, then this
bundle is said to have been Directly Revealed
Preferred to all other affordable bundles
x2
a
x1
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Weak Axiom of Revealed Preference (WARP)
No two different bundles, a and b, can be
directly revealed preferred to each other
That is, if at prices (p1,p2) bundle (a1,a2) is
chosen, and at prices (q1,q2) bundle (b1,b2) is
chosen, then it cannot be the case that
, b affordable when a chosen
, a affordable when b chosen
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Weak Axiom of Revealed Preference (WARP)
x2
Graphically, we cannot have
a
b
x1
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Indirect Revealed Preference
If a bundle a is directly revealed preferred to a
different bundle b, and bundle b is directly
revealed preferred to an alternative bundle c,
then we can say that bundle a is Indirectly
Revealed Preferred to bundle c.
x2
a
b
c
x1
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Strong Axiom of Revealed Preference (SARP)

• No two different bundles, a and c, can be either
  directly or indirectly revealed preferred to each
  other
• Note that both the strong and weak axioms of
  revealed preference require consumers to select a
  unique consumption bundle at any budget
  constraint

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Generalised Axiom of Revealed Preference(GARP)

• If bundle a is indirectly revealed preferred to
  bundle c then whenever c is purchased the
  purchase of a would have been more expensive,
  that is pca ? pcc
• This is a more general specification, as it
  allows the consumer to be indifferent between two
  affordable consumption bundles. Effectively
  allowing flat spots in the indifference curves

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Afriats Theorem

• For any finite number of price/consumption
  observations, there exists a continuous utility
  functions satisfying the condition that more is
  better, and with convex indifference curves which
  rationalises the data if and only if the
  observations satisfy GARP
• Here is Homo Economicus

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Question

• Is the utility function obtained by this process
  unique?
• NO!
• Utility functions are ordinal, so the same set of
  indifference curves may be generated by many
  different utility functions
• Practically, any finite number of observations
  may be explained by a number of different
  indifference maps, each generating different
  out-of-sample behaviour

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Index Numbers

• Consider a consumer who, in some base period, is
  observed consuming a bundle b when prices are pb
  and in some subsequent period is observed
  consuming bundle c when prices are pc
• Can anything be said about the welfare of this
  consumer over the interval?
• From revealed preference If pb.b gt pb.c then
  the consumer is worse off. If pc.c gt pc.b then
  the consumer is better off

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Indices
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Indices
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Consumer Price Index
The Consumer Price Index (CPI) measures the
proportional change in the cost of purchasing a
given bundle of commodities. Specifically,
The CPI is commonly compared with the growth in
nominal incomes to assess real income changes
Is this comparison justifiable?
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Consumer Price Index
If CPI lt M (ratio of incomes) then consumers are
better off. But if CPI gt M cannot conclude that
consumers are worse off.
x2
u0
Increasing incomes by CPI will generally improve
welfare
c
b
x1
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Readings

• Varian, Intermediate Microeconomics, chapter 7
• Varian, Microeconomic Analysis, chapter 8
• Samuelson P. (1948) Consumption theory in terms
  of revealed preference Econometrica, vol. 15,
  pp. 243-253.

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Next Time

• Welfare
• Consumer surplus
• Equivalent variation
• Compensating variation
• Slutsky vs Hicksian substitution