Consumer Comparative Statics — Market Demand
Lecture 3B
UNSW ECON5101
June 5, 2023
UNSW ECON5101 Consumer Comparative Statics — Market Demand June 5, 2023 1 / 20
Motivation
Elements of a decision problem:
What is feasible?
What is desirable?
We can now put them together to talk about choice. We will:
1 solve the consumer’s optimisation problem to derive their individual demand function.
(previously)
2 look at some comparative statics: how consumer demand changes as exogenous variables change.
(now)
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1 Consumer Comparative Statics
Elasticity
Hicks-Allen Decomposition
2 Market Demand
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The Exogenous Variables
We are interested in how quantity demanded of a good responds to changes in the following three
exogenous variables:
own price
other price(s)
income
For each, the obvious way to measure this is unit-dependent:
“If income increases by a dollar, how many more kilograms of rice does the consumer buy?”
This method does not allow us to measure sensitivities of goods to prices in a comparable way.
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Elasticity
The solution is elasticity, a ratio of percentage changes, which is unitless measure of demand
responsiveness.
ϵ =
%∆Q
%∆P
=
∆Q
Q
∆P
P
=
P
Q
∆Q
∆P
Since we are interested in infinitesimally small changes, we can write this as
ϵ =
P
Q
∂Q
∂P
UNSW ECON5101 Consumer Comparative Statics — Market Demand June 5, 2023 5 / 20
Own-price elasticity
Own-price elasticity: responsiveness of QD to a change in a good’s own price.
P1
Q1
∂Q1
∂P1
usually negative - Law of Demand
|ϵ11| < 1: inelastic - QD relatively unresponsive to price changes.
|ϵ11| > 1: elastic - QD relatively responsive to price changes.
If own-price elasticity is positive, it is possible that the consumer increases their consumption of
the good when its price goes up: it is a Giffen good.
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Cross-price Elasticity
Cross-price elasticity: responsiveness of QD to a change in another good’s price.
P2
Q1
∂Q1
∂P2
ϵ > 0: substitutes - increase in price of good 2 leads to an increase in consumption of good 1.
ϵ < 0: complements - increase in price of good 2 leads to a decrease in consumption of good 1.
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Income elasticity
Income elasticity: responsiveness of QD to a change in income.
I
Q1
∂Q1
∂I
If income elasticity is negative, it means the consumer decreases their consumption of the good when
their income goes up: the good is an inferior good.
If income elasticity is positive, it means the consumer increases their consumption of the good when
their income goes up: the good is a normal good.
η < 1: necessary good - as your income rises, the rate at which you consume more of the good
decreases.
η > 1: luxury good - as your income rises, the rate at which you consume more of the good
increases.
In a two-good economy, it cannot be that both goods are inferior goods.
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Example: Elasticity Computations
Recall that for the utility function u(x1, x2) = 3x
0.25
1 x
0.75
2 , that individual demand functions are given by
x1(p1, p2,m) =
m
4p1
, x2(p1, p2,m) =
3m
4p2
.
Find ϵ11, ϵ22, ϵ12, ϵ21, η1 and η2.
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The Hicks Allen Decomposition of a Price Effect
The Hicks-Allen Decomposition is a way to decompose the effect of a price change on quantity
demanded into its:
Income Effect: change in QD due to change in purchasing power.
Substitution Effect: change in QD due change in relative prices.
desmos
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If the good is a normal good, both the substitution and income effects are negative and reinforce each
other.
If the good is an inferior good, the income effect is positive but the negative substitution effect usually
dominates.
In the extreme case of a Giffen good, the positive income effect dominates the negative substitution
effect, causing a net increase in quantity demanded in response to an increase in price.
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Example: Hicks-Allen Decomposition
Elliot’s preferences are given by the below utility function, and prices and income are as specified.
u(x1, x2) = x1 + 2
√
x2. m = 10, p1 = 2, p2 = 1.
p1 then falls to 1. Decompose the price effect into its substitution and income effects using the
Hicks-Allen decomposition. desmos
We need to find the optimal bundles C,B, and A using the Tangency method.
Bundle C: Original optimal bundle.
MRS =
1
1√
x2
=
√
xC2 = 2 =
p1
p2
xC2 = 4.
B.C. =⇒ 2xC1 + xC2 = 2xC1 + 4 = 10
=⇒ xC1 = 3.
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Bundle A: New optimal bundle.
MRS =
p1
p2
⇐⇒
√
xA2 = 1 ⇐⇒ xA2 = 1
B.C. =⇒ xA1 + xA2 = xA1 + 1 = 10
=⇒ xA1 = 9.
Bundle B: Income adjusted bundle - needs to have the same price ratio as bundle A, and the same
utility as bundle C.
Given that the price ratio is the same as the case for bundle A, the Tangent condition tells us we have
xB2 = 1 again.
Bundle C has a utility of u(3, 4) = 3 + 2
√
4 = 7.
Hence, in bundle B, u(xB1 , 1) = x
B
1 + 2
√
1 = 7 =⇒ xB1 = 5
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The substitution effect is given by xB1 − xC1 = 5− 3 = 2.
The income effect is given by xA1 − xB1 = 9− 5 = 4.
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From Individual to Market Demand
So far, we have worked with demand functions of the form xi (p,m).
From a demand function, we can derive a demand curve xi (pi ) by fixing all other prices and income at
some level. Once we’re focusing on one good and one price (its own), we conventionally write demand
as Q(P).
If consumers are price-takers, and goods are rival and excludable, we can sum individual demands as
follows:
Consider how much each individual is willing to buy at each price level, and sum those quantities.
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Example: Market Demand 1
Consider three consumers whose individual demands are given respectively by: p = 30− qA,
p = 30− 2qB , and p = 30− 3qC .
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Example: Market Demand 2
Jeff’s demand is given by qJ(p) = 3− p and Anne’s is qA(p) = 1− p. Find the market demand for
these two individuals.
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Summary
You should be able to:
calculate own-price, cross-price and income elasticity of demand.
understand what characteristics goods may have given values for different types of elasticity.
use the Hicks-Allen decomposition to decompose a price effect into its income effect and
substitution effect.
aggregate individual demands into market demand for private goods.
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