chapter 3. * prerequisite: a binary relation r on x is said to be complete if xry or yrx for any...
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Chapter 3Chapter 3
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* Prerequisite: A binary relation R on X is said to be
Complete if xRy or yRx for any pair of x and y in X;
Reflexive if xRx for any x in X;
Transitive if xRy and yRz imply xRz.
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Rational agents and stable prefereRational agents and stable preferences nces
Bundle x is strictly preferred (s.p.), or weakly preferred (w.p.), or indifferent (ind.), to Bundle y.
(If x is w.p. to y and y is w.p. to x, we say x is indifferent to y.)
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Assumptions about PreferencesAssumptions about Preferences
Completeness: x is w.p. to y or y is w.p. to x for any pair of x and y.
Reflexivity: x is w.p. to x for any bundle x.
Transitivity: If x is w.p. to y and y is w.p. to z, then x is w.p. to z.
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The indifference sets, the indifference curves.
They cannot cross each other.
Fig.
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indifference curvesindifference curvesx2
x1
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Perfect substitutes and perfect complements. Goods, bads, and neutrals. Satiation. Figs
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Blue pencils
Red pencils
Indifference curves
Perfect Perfect substitutessubstitutes
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Perfect Perfect complementscomplements
Indifference curves
Left shoes
Right shoes
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Well-behaved preferences are monotonic (meaning more is better) and
convex (meaning average are preferred to extremes).
Figs
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x2
x1
Betterbundles(x1, x2)
MonotonicityMonotonicity
Betterbundles
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The marginal rate of substitution (MRS) measures the slope of the indifference curve.
MRS = d x2 / d x1, the marginal willingness to pay ( how much to give up of x2 to acquire one more of x1 ).
Usually negative. Fig
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Convex indifference curves exhibit a diminishing marginal rate of substitution.
Fig.
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x2
x1
ConvexityConvexity
Averagedbundle
(y1,y2)
(x1,x2)
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Chapter 4Chapter 4
(as a way to describe preferences)
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UtilitiesUtilities
Essential ordinal utilities,versus
convenient cardinal utility functions.
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Cardinal utility functions: u ( x ) ≥ u ( y ) if and only if bundle x is w.p. to bundle y.
The indifference curves are the projections of contours of
u = u ( x1, x2 ).
Fig.
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Utility functions are indifferent up to any strictly increasing transformation.
Constructing a utility function in the two-commodity case of well-behaved preferences:
Draw a diagonal line and label each indifference curve with how far it is from the origin.
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Examples of utility functionsExamples of utility functions u (x1, x2) = x1 x2 ;
u (x1, x2) = x12 x2
2 ;
u (x1, x2) = ax1 + bx2
(perfect substitutes); u (x1, x2) = min{ax1, bx2}
(perfect complements).
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Quasilinear preferences: All indifference curves are vertically (or h
orizontally) shifted copies of a single one, for example u (x1, x2) = v (x1) + x2 .
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Cobb-Douglas preferences:
u (x1, x2) = x1c x2
d , or
u (x1, x2) = x1ax2
1-a ;
and their log equivalents:
u (x1, x2) = c ln x + d ln x2 , or
u (x1, x2) = a ln x + (1– a) ln x2
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Cobb-DouglasCobb-Douglas
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MRS along an indifference curve.Derive MRS = – MU1 / MU2
by taking total differential along any indifference curve.
Marginal utilities
MU1 and MU2.
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MarginalMarginal analysis analysis
MM is the slope of the TM curve
AM is the slope of the ray from the origin to the point at the TM curve.
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500490
480 The demand curve
ReservationReservation priceprice
Number of apartment
From peoples’ reservation prices to the market demand curve.
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supply
Demand
PP
Q
EquilibriumEquilibrium
P*P*
Q*
E (P*,Q*)
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supply
Demand
pp
q
E
EquilibriumEquilibrium
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x2
x1
Budget lineBudget set
RationingRationing
R*
Marketopportunity
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MRSMRS
Indifferencecurve
Slope = dx2/dx1
x2
x1
dx2dx1