PREFERENCES AND RATIONALITY

MODELING CONSISTENCY IN WHAT WE WANT

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INTRODUCTION TO DECISION MAKING BY CONSUMERS

We want to study how individual agents (consumers and then firms) make optimal decisions

Two elements of any decision making problem what is feasible what is desirable

Last week we studied first element in a market setting Today we focus on the second one

we are interested in how individuals order the alternatives that are feasible to them

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ROAD MAP Are you crazy?

rational preference relations and their properties Whatever

indifference curves goods and bads, substitutes and complements

A taste for diversity monotonicity convexity

Trading off the marginal rate of substitution

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RATIONALITY IN ECONOMICS Behavioral postulate: a decision maker always chooses her

most preferred alternative from the set of feasible alternatives

this is clearly an approximation to the real world in various situations people not only make mistakes but exhibit

systematic biases

To model rational choice we must model the decision makers preferences systematic study of ranking of alternatives (ordinal ranking) construct a ranking of alternatives that makes sense

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PREFERENCE RELATIONS

Ordinal comparison between two different consumption bundles, x and y

(weak) preference: x is as at least as good as y, or equivalently x is weakly preferred to y

we are establishing a relation between two objects (consumption bundles x and y) that expresses a preference ranking

the preference relation is denoted by the symbol if x is weakly preferred to y, we write x y %

%

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PREFERENCE RELATIONS

Lets assume that bundle x is weakly preferred to y There are two possibilities

strict preference: x is strictly better than y indifference: x is exactly as preferred as is y

In the first case, bundle x is weakly preferred to y but bundle y is not weakly preferred to x we write x y and y x --- or equivalently x y

In the second case, bundle x is weakly preferred to y and bundle y is weakly preferred to x we write x y and y x --- or equivalently x y

% 6%

% %

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RATIONALITY ASSUMPTIONS Completeness: for any two bundles x and y it is always

possible to make the statement that either x is weakly preferred to y or y is weakly preferred to x it is always possible to compare two alternatives this rules out Sophies Choice situations (clip) completeness is a convenient but not essential

assumption; most of decision theory survives without it

Remark: reflexivity is a redundant assumption it is not necessary because it follows from completeness

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RATIONALITY ASSUMPTIONS Transitivity: if x is weakly preferred to y, and y is weakly

preferred to z, then it must be the case that x is weakly preferred to z

Transitivity is at the heart of rationality --- without it we dont have a theory transitivity rules out starving monkeys and money pumps

A preference relation is said to be rational if it is complete and transitive

Claim: if the preference relation is transitive, then the indifference and the strict preference relations are both transitive

x % y and y % z =) x % z

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INDIFFERENCE CURVES Fix a bundle x The set of all bundles equally preferred to x is the

indifference curve containing x: it contains all bundles y such that y is indifferent to x

An indifference curve is not always a curve

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I(x) =y 2 X : y x

INDIFFERENCE CURVES x2

x1

y = (y1, y2)

z = (z1, z2)

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x = (x1, x2)

I(x)

INDIFFERENCE CURVES x2

x1

y = (y1, y2)

z = (z1, z2)

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x = (x1, x2)

I(x)

INDIFFERENCE CURVES x2

x1

I1I2 I

All bundles in I1 are strictly preferred to all

bundles in I

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INDIFFERENCE CURVES x2

x1

I(x)

x

I(x)

WP(x) is the set of bundles weakly preferred to x

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INDIFFERENCE CURVES x2

x1

SP(x) is the set of bundles strictly preferred to x

It does not include I(x)

x

I(x)

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INDIFFERENCE CURVES CANNOT INTERSECT x2

x1

x y

z

I1 I2

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INDIFFERENCE CURVES CANNOT INTERSECT x2

x1

x y

z

I1 I2 Suppose otherwise:

From I1 we get x y, and from I2 we get x z Therefore y z

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SLOPES OF INDIFFERENCE CURVES Good 2

Good 1

two goods a negatively sloped indifference curve

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SLOPES OF INDIFFERENCE CURVES

one good and one bad a positively sloped indifference curve

Good 2

Good 1

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PERFECT SUBSTITUTES x2

x1 8

8

15

15 Slopes are constant at - 1

I2

I1

Bundles in I2 all have a total of 15 units and are strictly preferred to all bundles in I1, which have a total of only 8 units in them

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PERFECT COMPLEMENTS x2

x1

I1

45o

5

9

5 9

Each of (5,5), (5,9) and (9,5) bundles contains 5 pairs so each is equally preferred

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PERFECT COMPLEMENTS x2

x1

I1

45o

5

9

5 9

Each of (5,5), (5,9) and (9,5) bundles contains 5 pairs so each is equally preferred

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I2

INDIFFERENCE CURVES EXHIBITING SATIATION

x2

x1

Satiation (bliss) point

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INDIFFERENCE CURVES EXHIBITING SATIATION

x2

x1

Bet

ter

Satiation (bliss) point

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INDIFFERENCE CURVES EXHIBITING SATIATION

x2

x1

Bet

ter

Satiation (bliss) point

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WELL-BEHAVED PREFERENCES A rational preference relation is well-behaved if it is

monotone and convex

Monotonicity: more of any commodity is always preferred (i.e. no satiation and every commodity is a good)

Convexity: convex combinations (mixtures) of bundles are at least weakly preferred to the bundles themselves

example: the 50-50 mixture of the bundles x and y is z = .5 x + .5 y

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CONVEX PREFERENCES

x2

y2

x1 y1

x

y

z = x + y 2

is strictly preferred to both x and y

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x2

y2

x1 y1

x

y

z

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CONVEX PREFERENCES

CONVEX PREFERENCES

x2

y2

x1 y1

x

y

z

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Preferences are strictly convex when all mixtures z are strictly preferred to their component bundles x and y

NON-CONVEX PREFERENCES

x2

y2

x1 y1

z

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The convex combination z is less preferred than x or y

NON-CONVEX PREFERENCES

x2

y2

x1 y1

z

The convex combination z is less preferred than x or y

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Why is convexity important?

SLOPE OF INDIFFERENCE CURVES The slope of an indifference curve at a given consumption

bundle x is its marginal rate of substitution (MRS) at x It tells us the trade off in terms of preferences between

consumption of good 2 and consumption of good 1 to while keeping consumption at the indifference curve

how much of good 2 are we willing to forsake to increase consumption of good 1 (marginally)

equivalently, how much extra of good 2 do we need to consume if we decrease our consumption of good 1 (marginally)

How can a MRS be calculated?

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x2

x2

x1

MARGINAL RATE OF SUBSTITUTION

x1

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x

x2

x2

x1

MARGINAL RATE OF SUBSTITUTION

x1

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x

MRS(x) limx1!0

x2x1

=dx2dx1

x

THE MRS AND THE INDIFFERENCE CURVES Good 2

Good 1

two goods a negatively sloped

indifference curve MRS < 0

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THE MRS AND THE INDIFFERENCE CURVES

Bad

one good, one bad a positively sloped

indifference curve MRS > 0

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Good

MRS = - 5

MRS = - 0.5

MRS always increases with x1 (becomes less negative) if and only if preferences are strictly convex

THE MRS AND THE INDIFFERENCE CURVES

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x2

x1

Equivalently, the absolute value of the MRS is decreasing in x1 That is, convexity of preferences associated to diminishing marginal rate of substitution

THE MRS AND THE INDIFFERENCE CURVES

MRS = - 0.5

MRS = - 5

MRS decreases with x1 (becomes more negative) non-convex preferences

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x2

x1