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Chapter 5
Some Applications of Consumer Demand, and Welfare Analysis
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Price Sensitivity of Demand
Elasticity of demandPercentage change in demandFrom a given percentage change in price
2
%
%
.
qqq
p pp
q p
p q
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Price-Elasticity Demand Curves
Elastic demand, |ξ|>11% change in price
○ >1% change in quantity demanded
Inelastic demand, |ξ|<11% change in price
○ <1% change in quantity demanded
Unit elastic demand, |ξ|=11% change in price
○ =1% change in quantity demanded
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Elasticity along a linear demand curve
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Quantity 0
Price
A
Pmax
q= a-b.p
μ
|ξ|<1
|ξ|>1
P1 |ξ|=1
q p
p q
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Price-Elasticity Demand Curves
Perfectly inelastic demand curvePerfectly vertical demand curveZero quantity response to a price change
Perfectly elastic demand curveHorizontal demand curvePrice > p
○ Quantity = 0Price = p
○ Any quantity
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Perfectly elastic & perfectly inelastic demand curves
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Perfectly inelastic demand curve.With zero elasticity, the quantity demanded is constant as prices change.
Quantity 0
Price
D
(a)
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.
Quantity 0
Price
D
(b)
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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
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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.
Good 1 (x 1)0
Good 2 (x
2)
ef
B
B’
B”
B’”
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2. Ordinal utility property
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Regardless of the utility numbers assigned to the three indifference curves, the agent maximizes utility by choosing point e. Thus demand is unaffected
Good 1 (x 1)0
Good 2 (x
2)
B’
B
90(3)100(5)
120(8)
e
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From Individual to Market Demand
Market demand curveAggregate of individual demand curvesHorizontally add up individual demand curves
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Market demand from individual demand
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(a)
Person i
Quantity5 13
P1
P2
Price
(b)
Person j
Quantity10 20
Price
(c)
Person k
Quantity12 30
Price
(d)
Aggregate demand
Quantity27 63
Price
Di
Dj
Dk
The market demand curve D is the horizontal summation of the individual demand curves Di , Dj , and Dk .
D
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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
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1. Consumer Surplus
Consumer surplusNet gain to consumers measured as the
difference between the willingness to pay and the amount actually paid
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Quantity of cocaine demanded
0
Price
70
10
33.3
100
CS
1. Consumer Surplus
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.
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Measures of Consumer Gain/ Loss
Loss of consumer surplusDifference between
○ consumer surplus for price p○ consumer surplus for price p+∆p
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Change in consumer surplus
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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.
Good 1 (x 1) 0
Price
p
a
p+∆p
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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 functionMinimum 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
The Lagrangian
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uxxu
xpxpMinxx
),( 21
2211},{ 21
s.t.
)),((),,( 22121 xxuuxxuxxL 121 pp
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Derivation of an Expenditure function
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Suppose p1=$0.5 and P2=$1, What is the minimum level of income needed to bring the consumer to a utility level of u*?
Good 1 (x 1)0
Good 2 (x
2)
I1(u*)f
e
B110
15
7
20 B2 B3
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Measures of Consumer Gain/ Loss
Price-compensating variation in incomeAdditional income given to consumerAfter price changeSame utility (before price change)
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Price-compensating variation in income
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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
Good 1 (x 1)0
Good 2
I1
I2
f
ed
B”
Z
B’
B
p
Price-compensating variation (in income)
Suppose p1=$1 and P2=$1. If P2 increases to $2, How much extra income is needed to compensate the consumer?
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Price-Compensating Variations andExpenditure Functions
Prices: p1, p2
Utility level: u*Expenditure: E=E(p1,p2,u*)
Increase in p1 to p1+ϵExpenditure: E’=E(p1+ϵ,p2,u*)
Price-compensating variation = E’-E== E(p1+ϵ,p2,u*) - E(p1,p2,u*)
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