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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
1 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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)
2 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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.
3 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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]
4 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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.
5 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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
6 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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.
7 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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
8 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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.
9 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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.
10 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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
11 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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
12 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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.
13 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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
14 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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).
15 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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.
16 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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.
17 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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 ′
18 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
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)
19 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
The WARP and the Law of Demand
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.
20 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
The WARP and the Law of Demand
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
21 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
The WARP and the Law of Demand
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).
22 / 23
Microeconomics IChapter Two
S. Farshad Fatemi
Introduction
Demand Function
ComparativeStatics
The WARP andthe Law ofDemand
The WARP and the Law of Demand
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).
23 / 23
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