week 02 - part a
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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
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INDIFFERENCE CURVES x2
x1
y = (y1, y2)
z = (z1, z2)
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x = (x1, x2)
I(x)
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INDIFFERENCE CURVES x2
x1
y = (y1, y2)
z = (z1, z2)
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x = (x1, x2)
I(x)
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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
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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
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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
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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
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NON-CONVEX PREFERENCES
x2
y2
x1 y1
z
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The convex combination z is less preferred than x or y
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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?
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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
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x2
x2
x1
MARGINAL RATE OF SUBSTITUTION
x1
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x
MRS(x) limx1!0
x2x1
=dx2dx1
x
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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
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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
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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
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