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