week 2 lecture b - ch 7,8 with notes

7/24/2019 Week 2 Lecture B - Ch 7,8 With Notes

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Microeconomics ,1CK90

Week 2 Lecture B


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Revealed Preference

Chapter 7

/ School of Industrial Engineering PAGE 1


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Revealed preferences

In previous chapters we calculated the choice of the

consumer based on his preferences (modeled via the

utility function) and his budget line.

Here we reverse this process: we use the consumers

choices to discover information about his preferences.

We assume that preferences are well-behaved and in

fact, strictly convex (so that the optimal choice is unique)



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Principle of direct revealed preference

Let (,

) be the chosen bundle with prices 1 and 2.

It must be such that x

=

Let (1, 2) be another bundle which is affordable at

these prices,

It must be such that 11

Therefore,

.

The consumer chose (

,

) over (1, 2), we say that(1, 2

) is directly revealed preferred to(1, 2), and

this implies that

(,

) (1, 2)



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Recovering Indifference Curves

2

1

X

X is the optimal bundle for the blue budget line

To which bundles is X directly

revealed preferred to?


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2

1

X

Y

X is the optimal bundle for the blue budget lineY is the optimal bundle for the green budget line

To which bundles is Y directly

revealed preferred to?

To which bundles is X indirectly

revealed preferred to (but not

directly)?


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Principle of indirect revealed preference

If (,

) is directly revealed preferred to (1, 2)

And (1, 2) is directly revealed preferred to (1, 2)

Then (,

) is indirectly revealed preferred to (1, 2)

and this implies that

(,

) (1, 2)

by transitivity of the relation.



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2

1

X

Y


Z is the optimal bundle for the red budget line

To which other bundles is X

indirectly revealed preferredto?

Z


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2

1

X

Y



A is the optimal bundle for the purple budget line

To which other bundles is X

indirectly revealed preferredto?

ZA


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2

1

X

Y




What bundles are directly

revealed preferred to X? What bundles are preferred to X?

ZA


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2

1

X

Y




B is the optimal bundle for the dark blue budget line

What (new) bundles

are preferred to X?

ZA

B


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2

1

X

Y




B is the optimal bundle for the dark blue budget line

Where could the indifference

curve that goes through X be?

ZA

B


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Preferences consistent with maximizing

model

2

1

X

Y


What about this consumer?


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WARP (weak axiom of revealed preference)

If the bundle (1, 2) is directly revealed preferred to the

bundle (1, 2) then it cannot happen that (1, 2) is

directly revealed preferred to (1, 2).

If it happens anyway, we say that either:

This consumer is not maximizing, i.e., not choosing the best

bundle he/she can afford

Or some aspects of the choice problem changed between the

two lines (e.g., tastes changed)



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SARP (strong axiom of revealed preference)


If the bundle (1, 2) is revealed preferred (directly or

indirectly) to the bundle (1, 2) then it cannot happen that

(1, 2) is revealed preferred (directly or indirectly) to

(1, 2).

SARP is a necessary and sufficient condition for the

observed choices to be compatible with the economic

model of consumer choice.


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Example

Observation p1 p2 x1 x2

1 1 2 3 4

2 2 1 4 3

3 2 2 3 3


Are observationsconsistent with

WARP/SARP?


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Example



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Index numbers

We examine bundles in two time periods

is the base period: (,

) are the prices and ( ,

) is

the chosen bundle

is some other time: ( ,

) are the prices and ( ,

) is

the chosen bundle

We calculate indices to compare how consumption has

changed



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Index numbers: quantity indices

Paasche quantity index: =

+

+

What if > 1?

Laspeyres quantity index: =

+

+

What if < 1?



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Index numbers: price indices

Paasche price index:

=

Laspeyres quantity index:

=



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Slutsky equation

Chapter 8



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Effect of a change in price

We saw in Chapter 6 that when the price of a good

increases, the demand for this good

decreases if it is an ordinary good

Increases if it is a Giffen good.

In this chapter we decompose this effect into two effects:

A substitution effect: demand changes due to the change

in the rate of exchange between the two goods

An income effect: demand changes due to the change in

purchasing power



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Substitution and income effects

Suppose 1 decreases to


1

X Z

What happens to the quantity

consumed of good 1?

2


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Add a pivoted line which goes through

X and is parallel to the new budget line

This new budget line corresponds to the

same purchasing power as X but with the

same relative prices as the final budgetline


1

XZ

2


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What is the income that corresponds to the pivoted line?



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x1

XZ

Find the optimal bundle for the pivoted

budget linex2


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x1

XZ

Y

From X to Y is the substitution effect

From Y to Z is the income effectx2


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The substitution effect is:

The income effect is:

The sum of the two effects is the total change indemand:


=

=

=

1=


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Slutsky: absolute and marginal change

So, far: absolute changes: changes by

Denote original input and optimum by , &

From a marginal point of view (derivative):

Consider Slutsky demand function

1, 2, 1, 2 = 1, 2, 22 . Then



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Slutsky: absolute and marginal change

/ S h l f I d t i l E i i

week 2 lecture b - ch 7,8 with notes

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