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Chapter 5
Some Applications of Consumer Demand, and
Welfare Analysis
Price Sensitivity of Demand

Elasticity of demand
 Percentage change in demand
 From a given percentage change in price
 q 


%q  q 


%p  p 
 p 


 q   p 

 . 
 p   q 
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Price-Elasticity Demand Curves

Elastic demand, |ξ|>1
 1% change in price
○ >1% change in quantity demanded

Inelastic demand, |ξ|<1
 1% change in price
○ <1% change in quantity demanded

Unit elastic demand, |ξ|=1
 1% change in price
○ =1% change in quantity demanded
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Elasticity along a linear demand
curve
Price
Pmax
q= a-b.p
|ξ|>1
P1
q p

p q
μ |ξ|=1
|ξ|<1
0
A
Quantity
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Price-Elasticity Demand Curves

Perfectly inelastic demand curve
 Perfectly vertical demand curve
 Zero quantity response to a price change

Perfectly elastic demand curve
 Horizontal demand curve
 Price > p
○ Quantity = 0
 Price = p
○ Any quantity
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Perfectly elastic & perfectly inelastic
demand
curves
(a)
Price
Price
(b)
D
D
0
Quantity
Perfectly inelastic demand curve.
With zero elasticity, the quantity
demanded is constant as prices
change.
0
Quantity
Perfectly elastic demand curve.
With infinite elasticity, the quantity
demanded would be infinite for any price
below p and zero for any price above p.
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Properties of Demand
Functions
1.
Price and income multiplication by the
same factor leaves demand unaffected
 “No money illusion property”
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1. No Money Illusion Property
Good 2 (x 2)
B’
Multiplying all prices by the same
factor shifts the budget line from BB’
to B’”B”. Multiplying prices and the
agent’s income by the same factor
has no effect on the budget line.
B’”
e
f
0
B”
B
Good 1 (x 1)
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2. Ordinal utility property
Good 2 (x 2)
B
Regardless of the utility
numbers assigned to the three
indifference curves, the agent
maximizes utility by choosing
point e. Thus demand is
unaffected
e
120(8)
100(5)
90(3)
0
B’
Good 1 (x 1)
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From Individual to Market
Demand

Market demand curve
 Aggregate of individual demand curves
 Horizontally add up individual demand curves
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Market demand from individual
demand
(d)
(c)
(b)
(a)
Person i
Price
Person j
Price
Aggregate demand
Price
Person k
Price
P1
P2
Di
5 13
Quantity
Dk
Dj
10 20
Quantity
12
30
Quantity
D
27
63
Quantity
The market demand curve D is the horizontal summation of the individual
demand curves Di , Dj , and Dk .
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Welfare Measures

The welfare effects of price increase can be
assessed using
 Demand curve:
○ Loss in consumer surplus
 Consumer choice model:
○ Price compensating variation
1. Consumer Surplus
Consumer surplus
 Net gain to consumers measured as the
difference between the willingness to pay and the
amount actually paid
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1. Consumer Surplus
Price
33.3
CS
Consumer surplus.
The area under the demand
curve and above the price
measures the agent’s total
willingness to pay for the
quantity of the good she is
consuming minus the amount
she must pay.
10
0
70
100
Quantity of
cocaine
demanded
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Measures of Consumer Gain/
Loss

Loss of consumer surplus
 Difference between
○ consumer surplus for price p
○ consumer surplus for price p+∆p
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Change in consumer surplus
Price
a
p+∆p
p
0
Good 1 (x 1)
When the price increases, the change in the area under the demand
curve and above the price measures the welfare loss caused by the price
change.
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2. Price-compensating variation
in income
Price-compensating variation in income
measures the compensation needed due to an
increase in price
 To understand the price-compensating variation
in income we first introduce the expenditure
function
 Expenditure function

 Minimum income/expenditure amount (E)
 To achieve a predetermined utility (u)
 At given prices (p1,p2)

E=E(p1,p2,u)
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The Expenditure Function

The problem
Min p1 x1  p2 x2
{ x1 , x2 }
s.t. u ( x1 , x2 )  u

The Lagrangian
L( x1, x2 , u)  p1x1  p2 x2  (u  u( x1, x2 ))
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Derivation of an Expenditure
function
Suppose p1=$0.5 and P2=$1,
Good 2 (x 2)
17
15
What is the minimum level of
income needed to bring the
consumer to a utility level of u*?
e
f
7
0
I1(u*)
10
20
B1 B2
B3
Good 1 (x 1)
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Measures of Consumer Gain/
Loss

Price-compensating variation in income
 Additional income given to consumer
 After price change
 Same utility (before price change)
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Price-compensating variation in
incomeGood 2
Suppose p1=$1 and P2=$1. If P2
Price
-compensating
variation
(in income)
increases to $2, How much extra
income is needed to compensate
the consumer?
Z
B
f
d
e
I1
I2
p
0
B”
B’
Good 1 (x 1)
ZB is the amount of income that must be given to the agent after the price
increase in order to restore him to I1, the indifference curve he was on
before the price change
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Price-Compensating Variations and
Expenditure Functions

Prices: p1, p2
 Utility level: u*
 Expenditure: E=E(p1,p2,u*)

Increase in p1 to p1+ϵ
 Expenditure: E’=E(p1+ϵ,p2,u*)

Price-compensating variation = E’-E=
= E(p1+ϵ,p2,u*) - E(p1,p2,u*)
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