Neoclassical Theory of Consumption

Economics II: Microeconomics

V�E Praha

September 2009

Micro (V�E) Consumption theory 09/09 1 / 24

Economics II: Microeconomics

Consumers:

People.Households.Applications.

Firms:

Internal Organisation.Industrial Organisation.

Equilibrium:

Holds.Does not hold.

Micro (V�E) Consumption theory 09/09 2 / 24

Microeconomics

Consumers:

People.� NowHouseholds.Applications.

Firms:

Internal Organisation.Industrial Organisation.

Equilibrium:

Holds.Does not hold.

Micro (V�E) Consumption theory 09/09 3 / 24

Neoclassical theory of consumptionBasics

PEOPLE CHOOSE THE BEST THINGS THEY CAN AFFORD.

Micro (V�E) Consumption theory 09/09 4 / 24

Neoclassical theory of consumptionBasics

PEOPLE CHOOSE THE BEST THINGS THEY CAN AFFORD.

Micro (V�E) Consumption theory 09/09 5 / 24

Neoclassical theory of consumptionBasics

THE BEST THINGS:

De�nitionConsumtion bundle is a complete list of the goods & services that areinvolved in the choice problem that is investigated. (say, X and Υ).

FactNotation:Strict preference: X � ΥIndi¤erence: X � ΥAlternative notation:Weak preference: X � Υ

Micro (V�E) Consumption theory 09/09 6 / 24

Neoclassical theory of consumptionBasics

Relationship between �,�,and �

Lemma

if X � Υ and Υ � X then ?

if X � Υ but not Υ � X then ?

Micro (V�E) Consumption theory 09/09 7 / 24

Neoclassical theory of consumptionBasics

Relationship between �,�,and �

Lemma

if X � Υ and Υ � X then X � Υ

if X � Υ but not Υ � X then X � Υ

Micro (V�E) Consumption theory 09/09 8 / 24

Neoclassical theory of consumptionHomo-economicus

The economic agent (or economic human)

RationalEgoist (or self-interested)

Micro (V�E) Consumption theory 09/09 9 / 24

Neoclassical theory of consumptionHomo-economicus

The economic agent (or economic human)

Rational

Egoist (or self-interested)

Micro (V�E) Consumption theory 09/09 9 / 24

Neoclassical theory of consumptionHomo-economicus

The economic agent (or economic human)

RationalEgoist (or self-interested)

Micro (V�E) Consumption theory 09/09 9 / 24

Neoclassical theory of consumptionRationality

Axiom1st Axiom: Complete preferencesEither of X � Y

Y � XX � Y

jorjEither of X � Y

Y � Xboth

ExampleBuridan�s ass...

2nd Axiom: Re�exive preferencesX � X jorj X � X

Micro (V�E) Consumption theory 09/09 10 / 24

Neoclassical theory of consumptionRationality

Axiom

3rd Axiom: Transitive preferences

If X � Y and Y � Z then X � ZIf X � Y and Y � Z then X � Zor

If X � Y and Y � Z then X � Z

ExampleThe Dutch-booking...

Micro (V�E) Consumption theory 09/09 11 / 24

Indi¤erence curves

e = (15, 25)d = (10, 15)

e � c � aI 1 �indi¤erece curve

Micro (V�E) Consumption theory 09/09 12 / 24

Indi¤erence curves

I 0 : a � eI 1 : b � eQ : Do the axioms hold?

Question :(a) I 0 � I 1 � I 2;(b) I 0 � I 1 � I 2; or(c) I 2 � I 1 � I 0

Micro (V�E) Consumption theory 09/09 13 / 24

Indi¤erence curves

I 0 : a � eI 1 : b � eQ : Do the axioms hold?

PropositionIndi¤erence curvesrepresenting distinct levels ofpreference cannot cross.

Proof.Otherwise the transitivityaxiom is violated.

Micro (V�E) Consumption theory 09/09 14 / 24

Indi¤erence curves

I 0 : a � eI 1 : b � eQ : Do the axioms hold?

Question :(a) I 0 � I 1 � I 2;(b) I 0 � I 1 � I 2; or(c) I 2 � I 1 � I 0

Micro (V�E) Consumption theory 09/09 15 / 24

Indi¤erence curves

Psych. Assumptions

Axiom

4th Axiom: Insatiable(monotonic) preferences

If X � Y then X � YIf X > Y then X � Y

5th Axiom: Convex preferences

(w) If X � Y thenαX + (1� α)Y � Xwhere α 2 (0, 1)(s) If X � Y thenαX + (1� α)Y � X

Question :(a) I 0 � I 1 � I 2;(b) I 0 � I 1 � I 2; or(c) I 2 � I 1 � I 0

Micro (V�E) Consumption theory 09/09 16 / 24

Indi¤erence curves: �No-No-No�Cases

Micro (V�E) Consumption theory 09/09 17 / 24

Indi¤erence curves: Special Cases

Micro (V�E) Consumption theory 09/09 18 / 24

Marginal rate of substitution

De�nition

Marginal rate ofsubstitution (MRS) is therate at which the consumer isjust willing to substitute onegood for the other- MRS is the (absolute of the)slope of an indi¤erence curveat a particular point:

∆x2∆x1

ordx2dx1

Micro (V�E) Consumption theory 09/09 19 / 24

Diminishing Marginal Rate of Substitution

Micro (V�E) Consumption theory 09/09 20 / 24

Next lecture

Further studies in Neoclassical Theory!

Thank you!

Micro (V�E) Consumption theory 09/09 21 / 24

Utility

measure of happiness

cardinal utility

ordinal utility

De�nitionUtility function: A way ofassigning a number to everypossible consumption bundle,such taht more preferredbundles get assigned largernumbers.

Micro (V�E) Consumption theory 09/09 22 / 24

UtilityTheorem of Debreu

TheoremGiven the assumptions of Rationality and Monotonicity,9u (�) s.t. (x1, x2) � (y1, y2) () u (x1, x2) > u (y1, y2) .

Proof.Do not need for this course.

Micro (V�E) Consumption theory 09/09 23 / 24

UtilityMonotonic transforations

LemmaAny monotonic transformation of the original utility function is a utilityfunction representing the same preferences.

Proof.1. Suppose u (�) is the utility function representing the preferences �P .

(x1, x2) �P (y1, y2)() u (x1, x2) > u (y1, y2) (1)

2. And suppose that f (u) is a monotonic transformation of u (�) .

u (x1, x2) > u (y1, y2)() f (x1, x2) > f (y1, y2) (2)

3. From (1) and (2) follows:

(x1, x2) �P (y1, y2)() f (x1, x2) > f (y1, y2) (3)Micro (V�E) Consumption theory 09/09 24 / 24