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Part IIA, Paper 1 Consumer and Producer Theory. Lecture 4 Revealed Preferences and Consumer Welfare Flavio Toxvaerd. Today’s Outline. Leftovers from Lecture 2 Revealed preferences WARP, SARP, GARP Indices . Primal Approach. Dual Approach. Duality. Integrability problem. Solve. Solve.

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Part IIA, Paper 1 Consumer and Producer Theory

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Part iia paper 1 consumer and producer theory l.jpg

Part IIA, Paper 1Consumer and Producer Theory

Lecture 4

Revealed Preferences and Consumer Welfare

Flavio Toxvaerd


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Today’s 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

Roy’s Identity

Substitute into u(x,y)

Shephard’s 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

p1

a

b

Marshallian Demand

x1


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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 Preferredto 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 awould 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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Afriat’s 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 > pb.c then the consumer is worse off. If pc.c > pc.b then the consumer is better off


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Indices

Laspeyres Quantity Index

Paasche Quantity Index


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Indices

Laspeyres Quantity Index

Paasche Quantity Index


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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 < M (ratio of incomes) then consumers are better off.

But if CPI > 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


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