Download Lecture 5: Consumer Theory (cont`d)

Lecture 3: Consumer Theory (cont’d)
Topics 1.5
Properties of Demand
Keywords: Relative prices, real income
Substitution and income effects,
Slutsky equation, Law of Demand
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1.5.1 Relative Prices and real
income
Relative prices of good x in terms of good y =
units of good y foregone per unit of good x
acquired.
px / py = units of y / a unit of x
Example. px / py = 3 means that to get a unit of
x, we have to sacrifice 3 units of y. In other
words, price of x is 3 times higher than of y.
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1.5.1 Relative Prices and real
income
Real income: number of units of goods we could
get from our money income.
Reflect our purchasing power over a single good.
Nominal income / price of good x = units of x.
Marshallian demand is HD 0 in prices and income.
No money illusion
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1.5.1 Relative Prices and real
income
Theorem 1.10: Homogeneity and Budget
Balancedness
“Given u is continuous, st. increasing, and st.
quasiconcave, x(p,y) is HD zero in all prices and
income, and it satisfies budget balancedness, px=y
for all (p,y)”.
Idea: as MU is positive, you will use up your
money. For any (p,y), budget is binding.
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1.5.1 Relative Prices and real
income
As for Homogeneity, we can use price of one good
to be money unit. All things are compared to the
price of this good. We call this good as numeraire.
Say pn be numeraire; let t = 1/ pn
x(p,y) = x(tp,ty) = x ( p1 /pn , ... , pn-1 /pn , y / pn)
Demand depends on only n-1 relative prices, and
real income.
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1.5.2 Income and Substitution
Effects
How consumers respond to a relative price change.
Suppose the price of good 1 becomes cheaper, we
will buy more of good 1.
Very likely, but could this be only an answer?
Let talk about few concepts on
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Effect of a Price Change
Clothing
(units per
month)
Assume:
•I = $20
•PC = $2
•PF = $2, $1, $.50
10
A
6
U1
5
D
B
U3
4
Three separate
indifference curves
are tangent to
each budget line.
U2
4
12
20
Food (units
per month)
Effect of a Price Change
Price
of Food
Individual Demand relates
the quantity of a good that
a consumer will buy to the
price of that good.
E
$2.00
G
$1.00
Demand Curve
$.50
H
4
12
20
Food (units
per month)
Two Important Properties of Demand Curves
1) The level of utility that can be attained changes
as we move along the curve.
2)At every point on the demand curve, the
consumer is maximizing utility by satisfying
the condition that the MRS of food for
clothing equals the ratio of the prices of food
and clothing.
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Effects of Income Changes
Clothing
(units per
month)
Assume: Pf = $1
Pc = $2
I = $10, $20, $30
Income-Consumption
Curve
7
D
5
U2
B
3
U3
An increase in income,
with the prices fixed,
causes consumers to alter
their choice of
market basket.
U1
A
4
10
16
Food (units
per month)
Effects of Income Changes
Price
of
food
An increase in income,
from $10 to $20 to $30,
with the prices fixed,
shifts the consumer’s
demand curve to the right.
E
$1.00
G
H
D3
D2
D1
4
10
16
Food (units
per month)
Income Changes
An increase in income shifts the budget line to
the right, increasing consumption along the
income-consumption curve.
Simultaneously, the increase in income shifts the
demand curve to the right.
What else can shift the budget line? Changes in
income distribution, population, prices of
substitutes, preference, and price expectation
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1.5.2 Income and Substitution
Effects
A fall in the price of a good has two effects:
Substitution & Income
Substitution Effect
z Consumers will tend to buy more of the
good that has become relatively cheaper,
and less of the good that is now relatively
more expensive.
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1.5.2 Income and Substitution
Effects
A fall in the price of a good has two effects:
Substitution & Income
Income Effect
zConsumers experience an increase in real
purchasing power when the price of one
good falls.
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1.5.2 Income and Substitution
Effects
Substitution Effect
The substitution effect is the change in an
item’s consumption associated with a change
in the price of the item, with the level of utility
held constant.
When the price of an item declines, the
substitution effect always leads to an increase
in the quantity of the item demanded.
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1.5.2 Income and Substitution
Effects
Income Effect
The income effect is the change in an item’s
consumption brought about by the increase in
purchasing power, with the price of the item held
constant.
When a person’s income increases, the quantity
demanded for the product may increase or
decrease.
Even with inferior goods, the income effect is rarely
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large enough tomf620
outweigh
the
substitution
effect.
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Income and Substitution
effects: Normal Good
Clothing
(units per
month) R
When the price of food falls,
consumption increases by F1F2
as the consumer moves from A
to B.
The substitution effect,F1E,
(from point A to D), changes the
A
relative prices but keeps real income
(satisfaction) constant.
C1
D
B
C2
U2
Substitution
Effect
O
F1
Total Effect
The income effect, EF2,
( from D to B) keeps relative
prices constant but
increases purchasing power.
U1
E S
F2
T
Income Effect
Food (units
per month)
Income and Substitution
Effects: Inferior Good
Clothing
(units per
month) R
Since food is an
inferior good, the
income effect is
negative. However,
the substitution effect
is larger than the
income effect.
A
B
U2
D
Substitution
Effect
O
F1
U1
E S
Total Effect
F2
Income Effect
T
Food (units
per month)
p1
Decomposition of a price change
p10
p1
p11
X1(p1, p2o, yo)
x1h(p1, p2o, uo)
x1
SE
IE
1.5.2 Income and Substitution effect
Total price effect
= Substitution effect + Income effect
dx1 = ∂x1
+ ∂x ∂ I
dp1
∂p1 | u=u*
∂I ∂p1
Since ∂ I /∂p1 = x from budget equation. That is, when
price of good 1 increases by one Baht, you need more
of x Baht to feel the same after the change. This reflects
a loss of purchasing power, so we use –x to replace this
term.
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1.5.2 Income and Substitution effect
The term –x represents the amount of income
needed to keep u* at the original level.
Minus means change in purchasing power is in
the opposite direction to a price change.
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1.5.2 Income and Substitution effect
Theorem 1.11: Slutsky Equation
“ Let x(p,y), u* maximum utility at (p,y), then
∂x i (p,y) = ∂ x ih (p,u*) - xj (p,y) ∂x i(p,y)
∂p j
∂p j
∂y
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1.5.2 Income and Substitution effect
Theorem 1.11: Slutsky Equation
“ Let i=1 and j=1
∂x 1 (p,y) = ∂ x1h (p,u*) - x 1 (p,y) ∂x1(p,y)
∂p 1
∂p 1
∂y
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1.5.2 Income and Substitution effect
Theorem 1.11
“ Let i=1 and j=2
∂x1 (p,y) = ∂ x1h (p,u*) - x2 (p,y) ∂x1(p,y)
∂p2
∂p2
∂y
TE
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SE
IE
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Theorem 1.11 Slutsky Equation
Proof: xih (p, u*) = xi (p, e(p, u*) )
Dif. w.r.t pj and apply the chain rule
∂xih (p, u*) = ∂xi (p,e(p, u*)) + ∂ xi (p,e(p,u*)) ∂e(p,u*)
∂pj
∂pj
∂y
∂pj
Use the fact that u*=v(p,y), thus
e(p,u*)= e(p, v(p,y))= y ............(1)
And the last term is just the Hicksian demand
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Theorem 1.11 Slutsky Equation
∂e (p, u*) = xjh (p,u*) = xj h (p,v(p,y))
∂pj
The last term is just xj (p,y) ..............(2).
Substitute (1) and (2), we obtain
∂xih (p, u*) = ∂xi (p,y) + ∂ xi (p,y) xj (p,y) .....Q.E.D.
∂pj
∂pj
∂y
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Theorem 1.12 Negative own-SE
Let xih(p,u) be Hicksian demand for good i, then
∂xih (p, u*) ≤ 0, i = 1, ..., n.
∂p i
Proof: From Shepard’s lemma,
∂e (p, u) = xih (p,u) => ∂2e (p, u) = ∂ xih (p,u)
∂pi
∂pi2
∂pi
Since e is a concave function of p, thus all of its
second-order own
partial derivatives are nonpositive.
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Theorem 1.13 Law of Demand
A decrease in the own price of a normal good
will cause quantity demanded to increase.
If an own price decrease causes a decrease in
quantity demanded, the good must be inferior.
Proof: normal good, look at the Slutsky equation,
we know the SE is negative (THM 1.12) and the
IE is also negative (since ∂x/ ∂y > 0)
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Theorem 1.13 Law of Demand
Proof: inferior good, look at the Slutsky equation,
we know the SE is negative (THM 1.12) and the
IE is now positive (since ∂x/ ∂y < 0 by inferiors),
it is possible then IE dominates SE, thus the total
price effect becomes positive.
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Theorem 1.14 Symmetric
Substitution Terms
Let xh (p,u) , e(p,u) is c2, then
∂xih (p, u*) = ∂xjh (p, u*) i, j = 1, ..., n.
∂pj
∂pi
Proof.
∂2e (p, u) = ∂ xih (p,u)
∂pi ∂pj
∂pj
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Theorem 1.15 Negative Semidefinite
Substitution Matrix
Substitution Matrix contains all Hicksian Substitution
terms.
But this matrix is just the Hessian Matrix or secondorder partial derivative matrix of the expenditure
function.
∂ xih (p,u) = ∂2e (p, u)
∂pj
∂pi ∂pj
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Theorem 1.16 Symmetric and
Negative Semidefinite Slutsky Matrix
Replace elements in Substitution matrix using
Slutsky equation
This property and HD property of demand can
be used to test the consumer theory.
In empirical demand function, we can test if
these properties hold.
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1.5.3 Some Elasticity Relations
Price Elasticity of Demand
Recall: Price elasticity of demand measures the
percentage change in the quantity demanded
resulting from a one percent change in price.
ΔQ/Q ΔQ / ΔP
EP =
=
Q/P
ΔP/P
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1.5.3 Some Elasticity Relations
Income Elasticity of Demand
Recall: Income elasticity of demand measures
the percentage change in the quantity
demanded resulting from a one percent
change in income.
ηi = xi /xi
y / y
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1.5.3 Some Elasticity Relations
1. Budget constraint must hold with equality at
any given prices and income.
y = p1x1 (p,y) + p2x2 (p,y)
2. Engel Aggregation: Sum of income
elasticities weighted by its budget share =1.
s1 η1 + s2 η2 = 1 where s1=p1x1/y
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1.5.3 Some Elasticity Relations
3. Cournot Aggregation: weighted sum of crossprice elasticities of good j = - budget share of
good j.
s1 ∈1j + s2 ∈2j = -sj , j =1, 2
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Summary Theorem 1.10 – 1.17
All are implications from UM behavior.
1.10 Homogeneity tells how demand responds to an
equiproprotionate change in all prices and income.
1.10 Budget balancedness: income will be used up.
1.11 Slutsky gives us sign restrictions on systems of
demands on how it responds to a price change. In
addition, tell us how to separate the price effect into
SE and IE.
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Summary Theorem 1.10 – 1.17
1.12 Negative own SE effect (own price change)
1.13 Law of demand
1.14 Symmetric SE matrix (n x n) (Cross price change)
1.15 Negative Semidefinite SE matrix
1.16 Sym and neg semidef Slutsky matrix. (here we
allow for all prices and income change on the system of
Marshallian demands)
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Summary Theorem 1.10 – 1.17
1.17 Engel Aggregation tells us how the
quantities demanded across demand systems
hang together, as a response to an income
change and 1.18 to a single price change
(Cournot aggregation).
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Applications of demand theory
Marshallian demand, although observable, is not
good for welfare comparison, as v(p,y) is based
on ordinal concept, so we cannot compare utility
of Mr. A vs. of Mr. B.
Hicksian demand, although unobservable, is
good for welfare comparison. We can use e(p1,
u0) – e(p0, u0) across persons and compared.
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Applications of demand theory
What you can do when you know demand for your
product, and their responsiveness to price (price
Elasticity).
What are the determinants of price elasticity?
SE: more substitutes=>more elastic
IE: luxuries =>more elastic than necessities
Budget share: more share =>more elastic (not always
true.)Ex. salt is price inelastic because of no close
substitutes and necessities, rather than having a tiny
share of budget.
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Applications of demand theory
How about Budget share: more share =>more elastic.
This is not always true. Ex. salt is price inelastic
because of no close substitutes and necessities,
rather than having a tiny share of budget.
How about high-priced and low-prices goods: high
price=>more elastic. Contradiction, we can have a
demand curve with constant elasticity along price
ranges.
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Applications of demand theory
We can use to analyze the effects of
Gasoline price increase
Tariff on Thai exports by foreign countries.
Village-fund
Unemployment benefits
Food coupon
Quota on imports or exports
Producer taxes or consumer taxes (subsidies).
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