Demand and Slutsky equation
Varian: Intermediate Microeconomics, 8e, chapters 6 and 8
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In this lecture, you will learn
• how a rational consumer reacts to changes in prices and income
• how animals react to changes in prices and income
• what substitution effect and income effect is
• why substitution effect cannot be positive
• what follows from this for the shape of the demand curve
• how to solve global warming and prevent blackouts
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Demand
Demand function = relationship between optimal
quantity and prices and income:
x1 = x1(p1, p2, m)
x2 = x2(p1, p2, m)
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Demand
Demand function = relationship between optimal
quantity and prices and income:
x1 = x1(p1, p2, m)
x2 = x2(p1, p2, m)
Comparative statics in consumer theory –
what is the demand effect of changes in
• prices
• income
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Change in price
Price elasticity of demand ( ) = percentage change in quantity (x1)
divided by percentage change in price of the same good (p1):
=
∆x1/x1
∆p1/p1
=
∆x1
∆p1
·
p1
x1
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Change in price
Price elasticity of demand ( ) = percentage change in quantity (x1)
divided by percentage change in price of the same good (p1):
=
∆x1/x1
∆p1/p1
=
∆x1
∆p1
·
p1
x1
Dividing goods according to price elasticity:
• ordinary good – reduction in price increases quantity demanded:
∆x1
∆p1
< 0, < 0
• Giffen good – reduction in price reduces quantity demanded:
∆x1
∆p1
> 0, > 0
Examples: rice and wheat in China (Jensen a Miller, AER, 2008)
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Ordinary good
Good 1 in this graph is ordinary:
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Giffen good
Good 1 in this graph is Giffen good:
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Price consumption curve and demand curve
As price changes the optimal choice moves along the price consumption
curve (PCC) or the price offer curve (POC).
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Price consumption curve and demand curve
As price changes the optimal choice moves along the price consumption
curve (PCC) or the price offer curve (POC).
Demand curve = the relationship between the optimal choice
and a price, with income and the other price fixed.
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Examples of PCC and demand – perfect substitutes
Utility function u(x1, x2) = x1 + x2.
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Examples of PCC and demand – perfect substitutes
Utility function u(x1, x2) = x1 + x2.
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Examples of PCC and demand – perfect substitutes
Utility function u(x1, x2) = x1 + x2.
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Examples of PCC and demand – perfect substitutes
Utility function u(x1, x2) = x1 + x2. Demand for good 1:
x1 =



0 when p1 > p2
any number between 0 and m/p1 when p1 = p2
m/p1 when p1 < p2
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Examples of PCC and demand – perfect complements
Utility functionu(x1, x2) = min{x1, x2}.
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Examples of PCC and demand – perfect complements
Utility functionu(x1, x2) = min{x1, x2}. The consumer chooses
x1 = x2 = x. =⇒ PCC is a straight line.
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Examples of PCC and demand – perfect complements
Utility functionu(x1, x2) = min{x1, x2}. The consumer chooses
x1 = x2 = x. =⇒ PCC is a straight line. By substituting x into
the BL, we get the demand function:
p1x + p2x = m ⇐⇒ x1 = x2 = x =
m
p1 + p2
.
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Change in price of the other good
Cross elasticity of demand ( C ) = percentage change in quantity of
good 1 (x1) divided by percentage change in price of good 2 (p2):
C =
∆x1/x1
∆p2/p2
=
∆x1
∆p2
·
p2
x1
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Change in price of the other good
Cross elasticity of demand ( C ) = percentage change in quantity of
good 1 (x1) divided by percentage change in price of good 2 (p2):
C =
∆x1/x1
∆p2/p2
=
∆x1
∆p2
·
p2
x1
Dividing goods according to cross elasticity of demand:
• substitutes – rise in p2 increases demand for good 1 x1:
∆x1
∆p2
> 0, C > 0
• complements – rise in p2 reduces demand for good 1 x1:
∆x1
∆p2
< 0, C < 0
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Change in income
Income elasticity of demand ( I ) = percentage change in quantity
demanded of good (x1) divided by percentage change in income (m):
I =
∆x1/x1
∆m/m
=
∆x1
∆m
·
m
x1
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Change in income
Income elasticity of demand ( I ) = percentage change in quantity
demanded of good (x1) divided by percentage change in income (m):
I =
∆x1/x1
∆m/m
=
∆x1
∆m
·
m
x1
Dividing goods according to income elasticity of demand:
• normal good – rise in income increases demand:
∆x1
∆m
> 0, I > 0
• luxury good: I > 1
• necessary good: I ∈ (0, 1)
• inferior good – rise in income decreases demand:
∆x1
∆m
< 0, I < 0
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Normal good
Good 1 in this graph is normal:
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Inferior good
Good 1 in this graph is inferior:
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Income consumption curve and Engel curve
As income changes the optimal choice moves along the income
consumption curve (ICC) or income expansion path (IEP).
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Income consumption curve and Engel curve
As income changes the optimal choice moves along the income
consumption curve (ICC) or income expansion path (IEP).
Engel curve (EC) = the relationship between optimal choice and income,
with prices fixed.
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Examples of ICC and EC – perfect substitutes
Utility function u(x1, x2) = x1 + x2 a p1 < p2
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Examples of ICC and EC – perfect substitutes
Utility function u(x1, x2) = x1 + x2 a p1 < p2
Consumer buys only good 1 =⇒ ICC is x2 = 0.
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Examples of ICC and EC – perfect substitutes
Utility function u(x1, x2) = x1 + x2 a p1 < p2
Consumer buys only good 1 =⇒ ICC is x2 = 0.
Demand for good 1 is x1 = m/p1
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Examples of ICC and EC – perfect substitutes
Utility function u(x1, x2) = x1 + x2 a p1 < p2
Consumer buys only good 1 =⇒ ICC is x2 = 0.
Demand for good 1 is x1 = m/p1 =⇒ EC is m = p1x1.
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Examples of ICC and EC – perfect complements
Utility function u(x1, x2) = min{x1, x2} a p1 > 0 a p2 > 0
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Examples of ICC and EC – perfect complements
Utility function u(x1, x2) = min{x1, x2} a p1 > 0 a p2 > 0
Consumer buys equal amounts of both goods – ICC is x2 = x1.
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Examples of ICC and EC – perfect complements
Utility function u(x1, x2) = min{x1, x2} a p1 > 0 a p2 > 0
Consumer buys equal amounts of both goods – ICC is x2 = x1.
Demand for good 1 is x1 = m
p1+p2
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Examples of ICC and EC – perfect complements
Utility function u(x1, x2) = min{x1, x2} a p1 > 0 a p2 > 0
Consumer buys equal amounts of both goods – ICC is x2 = x1.
Demand for good 1 is x1 = m
p1+p2
=⇒ EC is m = (p1 + p2)x1.
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Homotetick´e preference
Homotetic preferences = is (x1, x2) (y1, y2), then for all t > 0 holds
that (tx1, tx2) (ty1, ty2).
Quantities demanded of goods change at the same ratio as income.
=⇒ ICC and EC are straight lines from the origin.
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Homotetick´e preference
Homotetic preferences = is (x1, x2) (y1, y2), then for all t > 0 holds
that (tx1, tx2) (ty1, ty2).
Quantities demanded of goods change at the same ratio as income.
=⇒ ICC and EC are straight lines from the origin.
Examples of homotetic preferences:
• perfect substitutes
• perfect complements
• Cobb-Douglas preferences
Homothetic preferences are not very realistic. If demand increases at
a different rate than income, we have luxury or necessary goods.
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Examples of ICC and EC – quasilinear preferences
Utility function u(x1, x2) = v(x1) + x2 – ICs are vertically parallel.
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Examples of ICC and EC – quasilinear preferences
Utility function u(x1, x2) = v(x1) + x2 – ICs are vertically parallel.
With increasing income, ICC is first horizontal and then vertical.
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Examples of ICC and EC – quasilinear preferences
Utility function u(x1, x2) = v(x1) + x2 – ICs are vertically parallel.
With increasing income, ICC is first horizontal and then vertical.
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Examples of ICC and EC – quasilinear preferences
Utility function u(x1, x2) = v(x1) + x2 – ICs are vertically parallel.
With increasing income, ICC is first horizontal and then vertical.
Engel curve is increasing and then vertical.
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EXAMPLE: Demand in animal kingdom
Kagel et al. (e.g. Econ. Inquiry, 1975) showed that rats and pigeons
behave according to the laws of demand.
Chen et al. (JPE, 2006) found out that capuchin monkeys
• have stable preferences
• react rationally to changes in prices and income – their behavior
follows the axioms of revealed preference (GARP)
• under uncertainty behave irrationally (same as people) – reference
dependence, loss aversion
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Slutsky equation
We concentrate only on the effect of a change in price.
We divide the effect of a change in price in
• pivot – the Slutsky substitution effect
• shift – the income effect
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Slutsky equation
We concentrate only on the effect of a change in price.
We divide the effect of a change in price in
• pivot – the Slutsky substitution effect
• shift – the income effect
Substitution/income effect is
• positive when ↑ p ↑ x, or ↓ p ↓ x.
(same direction)
• negative when ↑ p ↓ x, or ↓ p ↑ x.
(opposite direction)
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Pivoted budget line
The pivoted BL has the same slope as the final BL, but the income is
compensated so that the original bundle (x1, x2) is just affordable.
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Pivoted budget line
The pivoted BL has the same slope as the final BL, but the income is
compensated so that the original bundle (x1, x2) is just affordable.
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Pivoted budget line (cont’d)
How to adjust the income so that (x1, x2) is just affordable?
The compensated income m = m + ∆m, where
∆m = x1∆p1.
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Pivoted budget line (cont’d)
How to adjust the income so that (x1, x2) is just affordable?
The compensated income m = m + ∆m, where
∆m = x1∆p1.
Example:
At my original income I consume 20 chocolates.
The price of chocolate increases from 20 to 30 CZK.
What should be the income compensation ∆m?
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Pivoted budget line (cont’d)
How to adjust the income so that (x1, x2) is just affordable?
The compensated income m = m + ∆m, where
∆m = x1∆p1.
Example:
At my original income I consume 20 chocolates.
The price of chocolate increases from 20 to 30 CZK.
What should be the income compensation ∆m?
The chocolates are just affordable if my income increases by
∆m = 20 × 10 = 200 CZK.
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The Slutsky substitution effect
Substitution effect (SE) measures how demand changes when we
change prices, keeping the purchasing power fixed (pivot):
∆xs
1 = x1(p1, m ) − x1(p1, m)
SE isolates the pure effect from changing the relative prices.
Sometimes called the change in compensated demand:
Consumer’s income is compensated for a change in price by ∆m.
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Slutsky substitution effect (graph)
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Slutsky substitution effect (graph)
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Sign of the substitution effect
Revealed preference – at the original price the consumer revealed,
that he prefers the bundle X to all affordable bundles (green area).
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Sign of the substitution effect
For a utility-maximizing consumer the bundle X is revealed
preferred to all bundles in the green part of the pivoted BL
=⇒ the substitution effect is never positive.
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The income effect
Income effect (IE) measures the change in demand the income changes
from m to m and prices remain fixed at (p1, p2) (shift):
∆xn
1 = x1(p1, m) − x1(p1, m )
IE isolates the pure effect of a change in purchasing power due to a price
change.
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The income effect
Income effect (IE) measures the change in demand the income changes
from m to m and prices remain fixed at (p1, p2) (shift):
∆xn
1 = x1(p1, m) − x1(p1, m )
IE isolates the pure effect of a change in purchasing power due to a price
change.
Income effect is
• negative for normal goods (e.g. ↑ p =⇒ ↓ m =⇒ ↓ x)
• positive for inferior goods (e.g. ↑ p =⇒ ↓ m =⇒ ↑ x)
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The substitution and income effect (graph)
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The substitution and income effect (graph)
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The substitution and income effect (graph)
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The Slutsky identity
The Slutsky identity – the total change in demand TE = SE + IE:
∆x1 = ∆xs
1 + ∆xn
1
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The Slutsky identity
The Slutsky identity – the total change in demand TE = SE + IE:
∆x1 = ∆xs
1 + ∆xn
1
From Slutsky identity and the fact that SE is non-positive follows:
Division of goods according to the effect of
income price
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The Slutsky identity
The Slutsky identity – the total change in demand TE = SE + IE:
∆x1 = ∆xs
1 + ∆xn
1
From Slutsky identity and the fact that SE is non-positive follows:
Division of goods according to the effect of
income price
inferior good (positive IE) ← Giffen good (positive TE)
Giffen good must be inferior (plus: |IE+
| > |SE0−
|).
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The Slutsky identity
The Slutsky identity – the total change in demand TE = SE + IE:
∆x1 = ∆xs
1 + ∆xn
1
From Slutsky identity and the fact that SE is non-positive follows:
Division of goods according to the effect of
income price
inferior good (positive IE) ← Giffen good (positive TE)
inferior good (positive IE) ordinary good (negative TE)
Giffen good must be inferior (plus: |IE+
| > |SE0−
|).
Inferior good can be a Giffen or an ordinary good.
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The Slutsky identity
The Slutsky identity – the total change in demand TE = SE + IE:
∆x1 = ∆xs
1 + ∆xn
1
From Slutsky identity and the fact that SE is non-positive follows:
Division of goods according to the effect of
income price
inferior good (positive IE) ← Giffen good (positive TE)
inferior good (positive IE) ordinary good (negative TE)
normal good (negative IE) ordinary good (negative TE)
Giffen good must be inferior (plus: |IE+
| > |SE0−
|).
Inferior good can be a Giffen or an ordinary good.
Ordinary good can be a inferior or a normal good.
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The Slutsky identity
The Slutsky identity – the total change in demand TE = SE + IE:
∆x1 = ∆xs
1 + ∆xn
1
From Slutsky identity and the fact that SE is non-positive follows:
Division of goods according to the effect of
income price
inferior good (positive IE) ← Giffen good (positive TE)
inferior good (positive IE) ordinary good (negative TE)
normal good (negative IE) → ordinary good (negative TE)
Giffen good must be inferior (plus: |IE+
| > |SE0−
|).
Inferior good can be a Giffen or an ordinary good.
Ordinary good can be a inferior or a normal good.
Normal good must be a ordinary good (the law of demand).
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SE a IE for inferior good (graph)
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The law of demand
The law of demand: If the demand for a good increases when income
increases (normal good), then the demand for that good
must decrease when its price increases.
Reasoning:
1 A normal good must be ordinary (TE−= SE0−+ IE−).
2 An ordinary good has a decreasing demand curve.
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SE and IE – perfect complements
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SE and IE – perfect complements
TE = IE, SE is always 0.
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Example – SE and IE – perfect complements
Utility function: u(x1, x2) = min{2x1, x2}
Income: m = 60
Original prices: (p1, p2) = (1, 1)
Price of good 1 increases to p1 = 2
What is the SE and IE of this change in price?
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Example – SE and IE – perfect complements
Utility function: u(x1, x2) = min{2x1, x2}
Income: m = 60
Original prices: (p1, p2) = (1, 1)
Price of good 1 increases to p1 = 2
What is the SE and IE of this change in price?
Demand function for good 1:
x1 =
m
p1 + 2p2
Original demand: x1(m, p1) = 20
Compensated income: m = m + x1∆p1 = 60 + 20 = 80
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Example – SE and IE – perfect complements
Utility function: u(x1, x2) = min{2x1, x2}
Income: m = 60
Original prices: (p1, p2) = (1, 1)
Price of good 1 increases to p1 = 2
What is the SE and IE of this change in price?
Demand function for good 1:
x1 =
m
p1 + 2p2
Original demand: x1(m, p1) = 20
Compensated income: m = m + x1∆p1 = 60 + 20 = 80
Using the demand function for good 1 we calculate the SE and IE:
SE: ∆xs
1 = x1(p1, m ) − x1(p1, m) = 20 − 20 = 0
IE: ∆xn
1 = x1(p1, m) − x1(p1, m ) = 15 − 20 = −5
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SE and IE – perfect substitutes
Change in price → consumption of a different good:
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SE and IE – perfect substitutes
Change in price → consumption of a different good: TE = SE, IE = 0
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SE and IE – perfect substitutes (cont’d)
Change in price → consumption of the same good:
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SE and IE – perfect substitutes (cont’d)
Change in price → consumption of the same good: TE = IE, SE = 0
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SE and IE – quasilinear preferences
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SE and IE – quasilinear preferences
TE = SE, IE is always 0.
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CASE: Solving the problem of global warming
One solution to global warming:
• tax emissions of CO2
• return the tax revenue to consumers
Will the plan work?
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CASE: Solving the problem of global warming (cont’d)
The original BL: px + y = m
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CASE: Solving the problem of global warming (cont’d)
The original BL: px + y = m
The BL with a tax: (p + t)x + y = m + tx ⇐⇒ px + y = m
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CASE: Solving the problem of global warming (cont’d)
The original BL: px + y = m
The BL with a tax: (p + t)x + y = m + tx ⇐⇒ px + y = m
The bundle (x , y ) is affordable at the original BL, but the consumer
chooses the bundle (x, y). =⇒ The consumer is worse off.
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CASE: Solving the problem of global warming (cont’d)
The original BL: px + y = m
The BL with a tax: (p + t)x + y = m + tx
The bundle (x , y ) is affordable at the original BL, but the consumer
chooses the bundle (x, y). =⇒ The consumer is worse off.
If instead of the transfer, government reduces the income tax, this might
increase incentives to work so that people earn a higher income.
In this case consumers might be even better off.
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CASE: Solving the problem of global warming (graph)
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CASE: Solving the problem of global warming (graph)
The plan works: people want less of CO2 and more of other goods.
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APPLICATION: Setting prices in energy markets
Electricity production suffers from an extreme capacity problem: it is
relatively cheap to produce up to capacity, and impossible to produce more.
Building capacity is expensive =⇒ necessary to reduce use in peak
demand. The demand for electricity depends strongly on temperature
which is easy to predict.
Question: How to set up pricing system so that those users who are able
to cut back on their electricity use have incentive to do so.
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APPLICATION: Setting prices in energy markets (cont’d)
One possible solution: Real Time Pricing (RTP)
RTP: large industrial users are equipped with special meters that allow
the price of electricity to vary from minute to minute, depending on signals
sent from the electricity generating company.
Georgia Power Company – the largest RTP program in the world:
In 1999 it was able to reduce demand on high-price days by inducing
some large customers to cut their demand by as much as 60%.
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APPLICATION: Setting prices in energy markets (graph)
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What should you know?
• Demand is a function of prices and income.
• Ordinary good: ↑ p1 =⇒ ↓ x1
Giffen good: ↑ p1 =⇒ ↑ x1
PCC connects the optima at different p1
Demand curve: relationship between p1 and x1
• Substitutes: ↑ p2 =⇒ ↑ x1
Complements: ↑ p2 =⇒ ↓ x1
• Normal good: ↑ m =⇒ ↑ x1
Inferior good: ↑ m =⇒ ↓ x1
ICC connects optima at different m
Engel curve: relationship between m and x1
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What should you know? (cont’d)
• Slutsky equation: TE = SE + IE
• SE: demand effect of a change in relative prices
(at an original purchasing power)
• IE: demand effect of a change in purchasing
power (at constant new relative prices)
• IE or SE is positive if ↑ p ↑ x or ↓ p ↓ x.
IE or SE is negative if ↑ p ↓ x or ↓ p ↑ x.
• SE cannot be positive (revealed preference)
• Giffen goods are inferior: TE+= SE0−+ IE+
• Normal goods are ordinary: TE−= SE0−+ IE−
=⇒ Normal goods have a decreasing demand.
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