Transcript Chapter 3

Chapter 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
preferences
 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 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.
Fig.
They cannot cross each other.
indifference curves
x2
x1
Perfect
substitutes and
perfect complements.
Goods, bads, and neutrals.
Satiation.

Figs
Perfect substitutes
Blue pencils
Indifference curves
Red pencils
Perfect complements
Left shoes
Indifference
curves
Right shoes
Well-behaved
preferences are
monotonic (meaning more is
better) and
convex (meaning average are
preferred to extremes).

Figs
Monotonicity
x2
Better
bundles
(x1, x2)
Better
bundles
x1
 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.
Convexity
x2
(y1,y2)
Averaged
bundle
(x1,x2)
x1
Chapter 4
(as a way to describe preferences)
Utilities
Essential
versus
convenient
functions.
ordinal
cardinal
utilities,
utility

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 functions
u
(x1, x2) = x1 x2 ;
u (x1, x2) = x12 x22 ;
u (x1, x2) = ax1 + bx2
(perfect substitutes);
u (x1, x2) = min{ax1, bx2}
(perfect complements).
Quasilinear preferences:
All indifference curves are vertically (or
horizontally) shifted copies of a single
one, for example u (x1, x2) = v (x1) + x2 .
Cobb-Douglas
preferences:
u (x1, x2) = x1c x2d , or
a
1-a
u (x1, x2) = x1 x2 ;
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-Douglas
Marginal
utilities
MU1 and MU2.
MRS
along an indifference curve.
Derive MRS = – MU1 / MU2
by taking total differential along
any indifference curve.
Marginal 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.
Reservation
price
500
490
480
The demand curve
Number of apartment
From peoples’ reservation prices to
the market demand curve.
Equilibrium
P
P*
supply
E (P*,Q*)
Demand
Q*
Q
Equilibrium
p
supply
E
Demand
q
x2
Rationing
Budget line
Budget
Marketset
opportunity
R*
x1
MRS
x2
Indifference
curve
Slope = dx2/dx1
dx2
dx1
x1