Lecture 3: DemandNeoclassical Theory of Consumption

Economics II: Microeconomics

V�E Praha

October 2009

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Economics II: Microeconomics

Consumers:

People.Households.Applications.

Firms:

Internal Organisation.Industrial Organisation.

Equilibrium:

Holds.Does not hold.

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Microeconomics

Consumers:

People.� NowHouseholds.Applications.

Firms:

Internal Organisation.Industrial Organisation.

Equilibrium:

Holds.Does not hold.

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Neoclassical theory of consumptionBasics

PEOPLE CHOOSE THE BEST THINGS THEY CAN AFFORD.

PEOPLE CHOOSE THE MOST PREFERRED BUNDLE THEY CAN AFFORD.

Consumers CHOOSE THE MOST PREFERRED BUNDLE THEY CAN AFFORD.

De�nitionA consumer is an economic agent who makes a decision on consumption.

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Neoclassical theory of consumptionBasics: A¤ordable Bundles

FactThe slope of BC = - pzpB .

De�nitionBudget constraint:The bundles of goods thatcan be bought if the entirebudget is spent on thosegoods at given prices.

ExampleBudget constraint:p1x1 + p2x2 = m.

x2 =mp2|{z}

intercept

�p1p2|{z}

slope

� x1

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Budget ConstraintShifts in prices and income

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Neoclassical theory of consumptionBasics

CONSUMERS CHOOSE THE MOST PREFERRED BUNDLE FROM THEIRBUDGET SET.

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Neoclassical theory of consumptionConsumer�s choice

FactConsumers�choice is a comboof their preferences andbudget.

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Neoclassical theory of consumptionConsumer�s choice

FactConsumers�choice is a comboof their preferences andbudget.

LemmaConsumer chooses the bundlewhere the indiference curve istangent to the budget line.

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Neoclassical theory of consumptionConsumer�s choice

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Neoclassical theory of consumptionConsumer optimisation problem: Calculus approach

Problemmaxx1,x2

U (x1, x2) ,.

s.t. p1x1 + p2x2 = m

Problem (Lagrange function)

maxx1,x2,λ

L = U (x1, x2) + λ (m� p1x1 � p2x2)

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Neoclassical theory of consumptionConsumer optimisation problem: Calculus approach

Problemmaxx1,x2,λ

L = U (x1, x2) + λ (m� p1x1 � p2x2)

SolutionFirst order conditions:

∂L∂x1

= 0 :∂U (x1, x2)

∂x1� λp1 = 0 (1)

∂L∂x2

= 0 :∂U (x1, x2)

∂x2� λp2 = 0 (2)

∂L∂λ

= 0 : m� p1x1 � p2x2 = 0 (3)

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Neoclassical theory of consumptionConsumer optimisation problem (Cont�d)

Problem

SolutionFrom (1) and (2): �

MUx1MUx2

=

� ∂U (x1,x2)∂x1

∂U (x1,x2)∂x2

=p1p2

(4)

Equations (3) and (4) give the solution: x�1 (p1, p2,m) and x�2 (p1, p2,m)

De�nitionThe pair (x�1 , x

�2 ) is the optimal choice of the consumer.

De�nitionDemand function: x = x (p1, p2,m)

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Utility function and indi¤erence curves

The indi¤erence curve is the utility function on a �xed level.

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Consumer Optimisation Problem

Full derivative of the utilityfunction U (x1, x2) :

dU =∂U∂x1

� dx1 +∂U∂x2

� dx2(5)

FactThe indi¤erence curve is theutility function on a �xed level.

So (5) can be rewritten as:

∂U (x1,x2)∂x1

∂U (x1,x2)∂x2

= �dx2dx1

(6)

CorollaryFrom (4) and (6) we have:

MRS =p1p2

(7)

FactThe internal rate of change isequal to the external rate ofchange.

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Consumer Optimisation ProblemNecessary condition

MRS =p1p2

internal RCh = external RCh

slope IC = slope BC

bene�t of consumingx1 as opposed to x2

=opportunity cost ofx1 in terms of x2

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Next lecture

Further studies in Neoclassical Theory!

Thank you!

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Neoclassical theory of consumptionDemand function

De�nitionDemand:The quantity demanded ateach possible price.

Demand function:

x1 = x1 (p1, p̄2, m̄)

Comparative statics

Shifts in m.Shifts in p1.Shifts in p2.

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Neoclassical theory of consumptionDemand function

De�nitionDemand:The quantity demanded ateach possible price.

Demand function:

x1 = x1 (p1, p̄2, m̄)

Comparative statics

Shifts in m.Shifts in p1.Shifts in p2.

Micro (V�E) Consumption theory 10/09 18 / 28

Neoclassical theory of consumptionDemand function

De�nitionDemand:The quantity demanded ateach possible price.

Demand function:

x1 = x1 (p1, p̄2, m̄)

Comparative statics

Shifts in m.

Shifts in p1.Shifts in p2.

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Neoclassical theory of consumptionDemand function

De�nitionDemand:The quantity demanded ateach possible price.

Demand function:

x1 = x1 (p1, p̄2, m̄)

Comparative statics

Shifts in m.Shifts in p1.

Shifts in p2.

Micro (V�E) Consumption theory 10/09 18 / 28

Neoclassical theory of consumptionDemand function

De�nitionDemand:The quantity demanded ateach possible price.

Demand function:

x1 = x1 (p1, p̄2, m̄)

Comparative statics

Shifts in m.Shifts in p1.Shifts in p2.

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Income expansion path and Engel curve

Income expansion path

Bundles demanded atdi¤erent income levelsIncome-consumptioncurveincome o¤er curve

Engel curve

x1 = x1 (p̄1, p̄2,m)

Demand as a functionof income only!

Q: Shape of IEP and EC

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Engel curve for perfect substitute

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Income elasticity & Engel curve

De�nitionThe income elasticity of demand (or income elasticity) is the precentagechange in the quantity demanded in response to a given percentagechange in income.

εM =%∆x1%∆m

=∆x1x1

∆mm

=

x 01�x1x1

m 0�mm

=∂x1∂m

� mx1

Note:

εM 6=∆x1∆m

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Income elacticity & Engel curve

Normal good : ε > 0

Inferior good: ε < 0

Quasilinear: ε = 0

Luxury goods: ε > 1

Necessities: ε < 1

Homothetic: ε = 1

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Income elacticity & Engel curve

Normal good : ε > 0

Inferior good: ε < 0

Quasilinear: ε = 0

Luxury goods: ε > 1

Necessities: ε < 1

Homothetic: ε = 1

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Income elacticity & Engel curve

Normal good : ε > 0

Inferior good: ε < 0

Quasilinear: ε = 0

Luxury goods: ε > 1

Necessities: ε < 1

Homothetic: ε = 1

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Income elacticity & Engel curve

Normal good : ε > 0

Inferior good: ε < 0

Quasilinear: ε = 0

Luxury goods: ε > 1

Necessities: ε < 1

Homothetic: ε = 1

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Income elacticity & Engel curve

Normal good : ε > 0

Inferior good: ε < 0

Quasilinear: ε = 0

Luxury goods: ε > 1

Necessities: ε < 1

Homothetic: ε = 1

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Income elacticity & Engel curve

Normal good : ε > 0

Inferior good: ε < 0

Quasilinear: ε = 0

Luxury goods: ε > 1

Necessities: ε < 1

Homothetic: ε = 1

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Neoclassical theory of consumptionDemand function (revisited)

Demand function:

x1 = x1 (p1, p̄2, m̄)

Comparative statics

Shifts in m.Shifts in p1.Shifts in p2.

Relative price as well asincome change!

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Income and Substitution E¤ect

De�nitionThe change in demand due to the change in the rate of exchange betweenthe two goods is called substitution e¤ect. (changed own price, other pricesand utility constant).

De�nitionThe change in demand due to the change in purchasing power is calledincome e¤ect. (prices are hold constant)

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Income and Substitution E¤ect

Substitution e¤ect is always negative due to the �well-behaved�indi¤erence curves!

What about the direction of the income e¤ect?

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Income and Substitution E¤ect

Substitution e¤ect is always negative due to the �well-behaved�indi¤erence curves!What about the direction of the income e¤ect?

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Income and Substitution E¤ect

Gi¤en goods (Inferior)

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Price and cross-price changes

Own price change

Ordinary goods:∂x1∂p1

< 0

Gi¤en goods:∂x1∂p1

> 0

Cross price change

substitute (not perfect)∂x1∂p2

> 0

complement (not perfect)∂x1∂p2

< 0

Micro (V�E) Consumption theory 10/09 27 / 28

Price and cross-price changes

Own price change

Ordinary goods:∂x1∂p1

< 0

Gi¤en goods:∂x1∂p1

> 0

Cross price change

substitute (not perfect)∂x1∂p2

> 0

complement (not perfect)∂x1∂p2

< 0

Micro (V�E) Consumption theory 10/09 27 / 28

Price and cross-price changes

Own price change

Ordinary goods:∂x1∂p1

< 0

Gi¤en goods:∂x1∂p1

> 0

Cross price change

substitute (not perfect)∂x1∂p2

> 0

complement (not perfect)∂x1∂p2

< 0

Micro (V�E) Consumption theory 10/09 27 / 28

Price and cross-price changes

Own price change

Ordinary goods:∂x1∂p1

< 0

Gi¤en goods:∂x1∂p1

> 0

Cross price change

substitute (not perfect)∂x1∂p2

> 0

complement (not perfect)∂x1∂p2

< 0

Micro (V�E) Consumption theory 10/09 27 / 28

Price and cross-price changes

Own price change

Ordinary goods:∂x1∂p1

< 0

Gi¤en goods:∂x1∂p1

> 0

Cross price change

substitute (not perfect)∂x1∂p2

> 0

complement (not perfect)∂x1∂p2

< 0

Micro (V�E) Consumption theory 10/09 27 / 28

Price and cross-price changes

Own price change

Ordinary goods:∂x1∂p1

< 0

Gi¤en goods:∂x1∂p1

> 0

Cross price change

substitute (not perfect)∂x1∂p2

> 0

complement (not perfect)∂x1∂p2

< 0

Micro (V�E) Consumption theory 10/09 27 / 28

Next lecture

Further applications!

Thank you!

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