chapter 3. * prerequisite: a binary relation r on x is said to be complete if xry or yrx for any...
TRANSCRIPT
Chapter 3Chapter 3
* 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.
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.)
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.
The indifference sets, the indifference curves.
They cannot cross each other.
Fig.
indifference curvesindifference curvesx2
x1
Perfect substitutes and perfect complements. Goods, bads, and neutrals. Satiation. Figs
Blue pencils
Red pencils
Indifference curves
Perfect Perfect substitutessubstitutes
Perfect Perfect complementscomplements
Indifference curves
Left shoes
Right shoes
Well-behaved preferences are monotonic (meaning more is better) and
convex (meaning average are preferred to extremes).
Figs
x2
x1
Betterbundles(x1, x2)
MonotonicityMonotonicity
Betterbundles
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
Convex indifference curves exhibit a diminishing marginal rate of substitution.
Fig.
x2
x1
ConvexityConvexity
Averagedbundle
(y1,y2)
(x1,x2)
Chapter 4Chapter 4
(as a way to describe preferences)
UtilitiesUtilities
Essential ordinal utilities,versus
convenient cardinal utility functions.
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.
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.
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).
Quasilinear preferences: All indifference curves are vertically (or h
orizontally) shifted copies of a single one, for example u (x1, x2) = v (x1) + x2 .
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
Cobb-DouglasCobb-Douglas
MRS along an indifference curve.Derive MRS = – MU1 / MU2
by taking total differential along any indifference curve.
Marginal utilities
MU1 and MU2.
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.
500490
480 The demand curve
ReservationReservation priceprice
Number of apartment
From peoples’ reservation prices to the market demand curve.
supply
Demand
PP
Q
EquilibriumEquilibrium
P*P*
Q*
E (P*,Q*)
supply
Demand
pp
q
E
EquilibriumEquilibrium
x2
x1
Budget lineBudget set
RationingRationing
R*
Marketopportunity
MRSMRS
Indifferencecurve
Slope = dx2/dx1
x2
x1
dx2dx1