s. farshad fatemi microeconomics...

Microeconomics IChapter Two

S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics

The WARP andthe Law ofDemand

Microeconomics I44715 (1396-97 1st Term) - Group 1

Chapter TwoConsumer Choice

Dr. S. Farshad Fatemi

Graduate School of Management and EconomicsSharif University of Technology

Fall 2017

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


Introduction

In this chapter, we start our study of consumer demand in thecontext of a market economy.

Consumer is the most fundamental decision unit ofmicroeconomic theory.

Market economy is the setting in which the goods andservices are available for purchase at the known prices.(consumers are price takers and their individual decisions haveno effect on these prices)

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


IntroductionCommodities

Commodity: a good or service available for purchase in themarket.

Commodity Vector: a list of amounts of the differentcommodities. The number of commodities is considered to befinite (l = 1, 2, ..., L).

x =

x1

...xL

∈ RL

Time and location can be built into the definition of acommodity.In principle, the elements of x can obtain negative values aswell. Negative consumption of a commodity can be interpretedas the net outflow or sale for a consumer.

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


IntroductionConsumption Set

Consumption Set: is a subset of commodity space (X ∈ RL) whichincludes only the consumption bundles which an individual canconceivably consume considering the physical constraints imposed bythe environment.

To keep things straightforward, we continue with the simplestsort of consumption sets:

X = RL+

Here, RL+ is defined as non-negative then it includes zeros as

well.Then RL

+ can be written as:

RL+ = {x ∈ RL : xl ≥ 0 for l = 1, 2, ..., L}

RL+ is a convex set.

if x , x ′ ∈ RL+

then x ′′ = αx + (1− α)x ′ ∈ RL+ for ∀α ∈ [0, 1]

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


IntroductionBudget Set

Budget Set: Consumer’s limit on how much she can spend.

Two assumptions:

Principle of completeness (universality) of markets: Allcommodities are traded in the market at prices which arepublicly quoted.

p =

p1

...pL

∈ RL

Note: prices can be negative. But we always assume p � 0.

Price-taking assumption: the prices are not affected byconsumer’s choice.

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


IntroductionBudget Constraint

If w is the consumer’s wealth level, then the consumer’s budgetconstraint can be interpreted as:

p.x =L∑

l=1

plxl ≤ w

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


IntroductionThe Walrasian Budget Set

Combining this economic-affordability constraint with therequirement that x ∈ RL

+ :

Definition (MWG 2.D.1): The Walrasian budget set orCompetitive budget set is the set of all feasible consumption bundlesfor the consumer who faces market prices p and has wealth w :

Bp,w = {x ∈ RL+ : p.x ≤ w}

The upper boundary of the budget set is {x ∈ RL+ : p.x = w}

which is called the budget hyperplane (or budget line for L = 2).

The Walrasian budget set is a convex set. Note: Bp,w is convexbecause RL

+ is convex. With a more general X : Bp,w is convexas long as X is.

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


Demand Function

The Walrasian (/market/ordinary) demand correspondence assigns aset of chosen consumption bundles for each price-wealth pair:x(p,w).

If x(p,w) is single-valued then it is called demand function.

Definition (MWG 2.E.1): The Walrasian demand correspondencex(p,w) is homogeneous of degree zero if

x(αp, αw) = x(p,w) for ∀p,w & α > 0

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Introduction

Demand Function

ComparativeStatics


Demand Function

Definition (MWG 2.E.2): The Walrasian demand correspondencex(p,w) satisfies Walras’ law if

∀p � 0 & w > 0, we have p.x = w for ∀x ∈ x(p,w)

From now on, we assume x(p,w) is always single-valued,continuous, and differentiable.

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


Comparative Statics

How demand is affected by changes in the consumer’s wealth andprices.

Engel function: For fixed prices p̄, the function of wealthx(p̄,w) is called the consumer’s Engel function.

Wealth (income) expansion path: The image of Engelfunction in the commodity space.

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


Comparative StaticsWealth Effect

Wealth (income) effect: The change in quantity demanded as aresult of a change in wealth (income):

∂xl(p,w)

∂w∀(p,w)

Matrix interpretation:

Dwx(p,w) =

∂x1(p,w)

∂w∂x2(p,w)

∂w...

∂xL(p,w)∂w

∈ RL

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Introduction

Demand Function

ComparativeStatics


Comparative StaticsWealth Effect

Normal good: A commodity l is normal at (p,w) if

∂xl(p,w)

∂w≥ 0

Inferior good: A commodity l is inferior at (p,w) if

∂xl(p,w)

∂w< 0

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Introduction

Demand Function

ComparativeStatics


Comparative StaticsPrice Effect

Price effect: The change in quantity demanded as a result of achange in prices:

Definition of the Demand functionDefenition of the Offer Curve.

The price effect of the price of good k on the demand for goodl :

∂xl(p,w)

∂pk∀(p,w)

l = k : own price effect;l 6= k : cross price effect.

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Introduction

Demand Function

ComparativeStatics


Comparative StaticsGiffen Good

Giffen good: A commodity l is Giffen at (p,w) if:

∂xl(p,w)

∂pl> 0

The definition of a Giffen good involves an income (wealth)effect, as well.A Giffen good is always an inferior good (Why?).

Matrix interpretation:

Dpx(p,w) =

∂x1(p,w)

∂p1· · · ∂x1(p,w)

∂pL...

. . ....

∂xL(p,w)∂p1

· · · ∂xL(p,w)∂pL

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Introduction

Demand Function

ComparativeStatics


Comparative StaticsElasticities of Demand

Price elasticity of demand:

εlk(p,w) =

∂xl (p,w)xl (p,w)

∂pkpk

=pk

xl(p,w).∂xl(p,w)

∂pk

Elasticity is independent of the units unlike the derivative itself.

Luxury and Necessity goods (defined based on incomeelasticity).

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


Comparative StaticsElasticities of Demand

Proposition (MWG 2.E.1): If the Walrasian demand functionx(p,w) is homogeneous of degree zero, then

L∑k=1

∂xl(p,w)

∂pkpk +

∂xl(p,w)

∂ww = 0 for l = 1, ..., L and ∀(p,w)

orDpx(p,w)p + Dwx(p,w)w = 0 ∀(p,w)

or

L∑k=1

εlk(p,w) + εw (p,w) = 0 for l = 1, ..., L and ∀(p,w)

An equal percentage change in all prices and wealth leads to nochange in demand.

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Introduction

Demand Function

ComparativeStatics


Comparative StaticsCournot and Engel Aggregation

Proposition (MWG 2.E.2 & 2.E.3): If the Walrasian demandfunction x(p,w) satisfies the Walras’ law, then (Cournotaggregation)

L∑l=1

∂xl(p,w)

∂pkpl + xk(p,w) = 0 for k = 1, ..., L and ∀(p,w)

and (Engel aggregation)

L∑l=1

∂xl(p,w)

∂wpl = 1 ∀(p,w)

Cournot aggregation: The total expenditure cannot change inresponse to a change in prices.

Engel aggregation: The total expenditure must change by anamount equal to any wealth change.

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics


The WARP and the Law of Demand

Assume x(p,w) is single-valued, homogeneous of degree zero, andsatisfies Walras’ law.

Definition (WARP) (MWG 2.F.1): The Walrasian demandfunction x(p,w) satisfies the weak axiom of revealed preferences ifthis property holds for any two price-wealth situation (p,w) and(p′,w ′) :

p.x(p′,w ′) ≤ w and x(p′,w ′) 6= x(p,w)⇒ p′.x(p,w) > w ′

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Introduction

Demand Function

ComparativeStatics


The WARP and the Law of DemandSlutsky Wealth Compensation

Slutsky Wealth Compensation: If the consumer initially facesprice-wealth pair of (p,w) and chooses x(p,w), then the pricevector changes to p′; the Slutsky wealth compensation adjusts theconsumer’s wealth to make the initially chosen bundle as affordableas before (x(p,w) is still on the budget hyperplane):

w ′ = p′.x(p,w) or w ′ = w + (p′ − p).x(p,w)

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics



Proposition (MWG 2.F.1): Suppose that the Walrasian demandfunction x(p,w) is homogeneous of degree zero and satisfies Walras’law; then it satisfies WARP if and only if the following propertyholds:

For any compensated price change from initial situation (p,w) to anew one (p′, p′.x(p,w)) we have:

(p′ − p).[x(p′,w ′)− x(p,w)] ≤ 0 if x(p′,w ′) 6= x(p,w)

Students need to go through the proof in MWG

The proposition simply states that demand and price move inopposite directions.

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics



dx = Dpx(p,w)dp + Dwx(p,w)dw

by Slutsky wealth compensation dw = x(p,w).dp

dx = [Dpx(p,w) + Dwx(p,w)x(p,w)T ]dp

from the proposition we have dp.dx ≤ 0 , so

dp.[Dpx(p,w) + Dwx(p,w)x(p,w)T ]dp ≤ 0

the middle part is an L× L matrix:

S(p,w) =

s11(p,w) · · · s1L(p,w)...

. . ....

sL1(p,w) · · · sLL(p,w)

Slutsky (substitution) matrix

where

slk(p,w) =∂xl(p,w)

∂pk+∂xl(p,w)

∂wxk(p,w) substitution effect

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics



Proposition (MWG 2.F.2): If a differentiable Walrasian demandfunction x(p,w) is homogeneous of degree zero and satisfies Walras’law and the WARP, then at any (p,w), the Slutsky matrix satisfies

v .S(p,w)v ≤ 0 ∀v ∈ RL

A matrix satisfying this property is called negative semi-definite.

Then sll(p,w) ≤ 0 which means

The own substitution effect is always nonpositive.

Recall the definition of a Giffen good (∂xl (p,w)∂pl

> 0) then since

sll(p,w) =∂xl(p,w)

∂pl+∂xl(p,w)

∂wxl(p,w) ≤ 0

for a good to be Giffen it should be an inferior good (∂xl (p,w)∂w < 0).

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S. Farshad Fatemi

Introduction

Demand Function

ComparativeStatics



Proposition (MWG 2.F.3): Suppose that the Walrasiandemand function x(p,w) is differentiable, homogeneous ofdegree zero, and satisfies Walras’ law, then for any (p,w):

S(p,w)p = 0

Note: Two theories of demand (i. based on rational preferencemaximisation; ii. Based on assumption of homogeneity ofdegree zero, Walras’ law, and WARP) are not equivalent.

Why? We will come back to this in details in the next chapter,while discussing how demand is generated from preferences.

For now, they are different, because the Walrasian budget setsdo not include every possible budget set (in particular allbudgets formed by 2 or 3 commodity bundles).

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