Chapter Five
APPLYING CONSUMER THEORY
1
Topics
1. Deriving Demand Curves
2. Effects of Changes in Income
3. Effects of a price change: Income and Substitution
Effects
4. Cost-of-Living Adjustments
5. Labor Supply Curves
5-2
Deriving Demand Curve
We will see how an individual demand curve is derived
from the utility maximization problem.
•Price Changes
– Using the figures developed in the previous chapter, the
impact of a change in the price of food can be
illustrated using indifference curves
– For each price change, we can determine how much of
the good the individual would purchase given their
budget lines and indifference curves
3
Deriving Demand Curve
•To derive an individual demand curve from the
utility maximization problem, consider the following
situation.
•Initial price of food: $2
•Price of clothing : $2
•The consumer’s income: $20
•Given those prices and income, the consumer’s
utility maximizing consumption bundle is F=4 and
C=6.
4
Deriving Demand Curve
Clothing
Assume:
• I = $20
• PC = $2
• PF = $2, $1, $0.50
10
A
6
U1
5
Each price leads to
different amounts of
food purchased
D
B
4
U3
Figure 1
U2
4
12
20
Food (units
per month)
5
Deriving Demand Curve
By changing prices and
showing what the consumer
will purchase, we can create
a demand schedule and
demand curve for the
individual
From the example:
Demand Schedule
P
Q
$2.00
4
$1.00
12
$0.50
20
6
Deriving Demand Curve
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
Figure 2
$.50
H
4
12
20
Food (units
per month)
7
Demand Curves
The utility maximizing quantity as a function of prices and
income is called the “demand function”.
The utility maximizing quantity for food is a function of the price
of food, income, and possibly the price of clothing. In the same
way, the utility maximizing quantity for clothing is a function of
the prices of clothing, food and income.
The individual demand curve can be seen as the graph of the
demand function for the own price, holding other variables
constant.
◦ For example, the demand curve for food is the plot of quantity demanded
against the price of food, holding the price of clothing and income constant.
8
Effect of a Price Change
Price
of Food
When the price falls,
Pf /Pc & |MRS| also fall
E
$2.00
• E: Pf /Pc = 2/2 = 1 = |MRS|
• G: Pf /Pc = 1/2 = .5 = |MRS|
• H:Pf /Pc = .5/2 = .25 =|MRS|
G
$1.00
$.50
H
4
12
20
Demand Curve
Food (units
per month)
9
Demand Curves – Important
Properties
The level of utility that can be attained changes as
we move along the curve
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 (|MRS|=PF/PC)
10
Deriving the demand curve from utility
maximization problem
[Exercise]
Suppose the utility function for a consumer is given
by
U(x,y) = xy
The budget for this consumer is $100
Let Px and Py be the prices of x and y respectively.
Find the demand function (the equation for the
demand curves) for x and y.
11
Effect of a Price Change
Clothing
10
A
6
U1
5
D
B
4
U3
The Price-Consumption Curve
traces out the utility maximizing
market basket for each price of
food. Notice that when price of
food changes, the quantity
demanded for clothing can also
change. This is because, if price of
food changes, the purchasing
power of income changes. For
example, if price of food
decreases, the ability to purchase
both goods increases.
U2
4
12
20
Food (units
per month)
12
Substitutes & Complements
Whether two goods are substitutes or complements
can be inferred from the price-consumption curve.
If the price consumption curve is downward-sloping,
the two goods are considered substitutes.
If the price consumption curve is upward-sloping, the
two goods are considered complements
They could be both.
13
Consumer Theory and Income
Elasticities
Income Changes
◦ As already discussed changing income, with prices
fixed, causes consumers to change their market baskets
◦ Three ways in which we can represent the relationship
between Income and Quantity Demanded
14
Effects of Income Changes
Assume: Pf = $1, Pc = $2
I = $10, $20, $30
Clothing
(units per
month)
7
D
5
U3
An increase in income,
with the prices fixed,
causes consumers to alter
their choice of
market basket.
U2
B
3
U1
A
4
10
16
Food (units
per month)
15
Consumer Theory and Income
Elasticities
Income Changes
◦ The income-consumption curve traces out the utilitymaximizing combinations of food and clothing
associated with every income level
16
Consumer Theory and Income
Elasticities
Income Changes
◦ An increase in income shifts the budget line to the
right, increasing consumption along the incomeconsumption curve
◦ Simultaneously, the increase in income shifts the
demand curve to the right
17
Consumer Theory and Income
Elasticities
Clothing
(units per
month)
Income
Consumption
Curve
7
D
5
U2
U3
If income increases and
prices are held constant, then
budget line will shift outward
in a parallel fashion, and the
utility maximizing
combination of goods
changes. Then an incomeconsumption curve traces out
the utility-maximization
combinations of goods with
every income level.
B
3
U1
A
4
10
16
Food (units
per month)
18
Consumer Theory and Income
Elasticities
Price
of
food
E
$1.00
G
H
D3
D2
An increase in income,
from $10 to $20 to $30,
with the prices fixed,
shifts the consumer’s
demand curve to the
right as well.
Notice in this case,
income consumption
curve is upward sloping.
This means that
consumption of both
goods increases as
income increases.
D1
4
10
16
Food (units
per month)
19
Consumer Theory and Income
Elasticities
•Engel Curves
– A curve that shows the relationship between the
quantity demanded of a single good and income,
holding prices constant
– The Engel curve is derived from the income
consumption cure.
– If the good is a normal good, the Engel curve is
upward sloping
– If the good is an inferior good, the Engel curve is
downward sloping
20
Engel Curves
Income 30
($ per
month)
Engel curves slope
upward for
normal goods.
20
10
4
8
12
16
Food (units
per month)
21
Consumer Theory and Income
Elasticities.
Formally,
Q
%Q
Q Y
Q
x


%Y Y Y Q
Y
◦ where Y stands for income.
Example
◦ If a 1% increase in income results in a 3% decrease in
quantity demanded, the income elasticity of demand
is x = -3%/1% = -3.
5-22
Consumer Theory and Income
Elasticities
Normal goods versus inferior goods
oInferior goods: goods whose quantity demanded falls as
income increases.
oNormal goods: goods whose quantity demanded increases
as income increases. Among normal goods, we can also
distinguish “necessities” whose quantity demanded rises
at a slower rate than income, and “luxuries” whose
quantity demanded rises at a faster rate than income.
23
Consumer Theory and Income
Elasticities
Income Changes
◦ Shape of the ICC for two goods tells us the sign of the
income elasticities: whether income elasticities for
those goods are positive or negative
◦ When the income-consumption curve has a positive
slope:
◦ The quantity demanded increases with income
◦ The income elasticity of demand is positive
◦ The good is a normal good
24
Consumer Theory and Income
Elasticities
Income Changes
◦ When the income-consumption curve has a negative
slope:
◦ The quantity demanded decreases with income
◦ The income elasticity of demand is negative
◦ The good is an inferior good
25
Housing, Square feet per year
Income-Consumption Curves and Income
Elasticities
Food inferior,
housing normal
L2
ICC 1
a
Food normal,
housing normal
ICC 2
As income rises the
budget constraint shifts
to the right.
◦ The income elasticities
depend on….
◦ …where on the new budget constraint
the new optimal consumption bundle
will be
b
L1
e
c
ICC 3
Food normal,
housing inferior
I
Food, Pounds per year
5-26
When Gail was poor
and her income
increased..
(a) Indifference Curves and Budget Constraints
All other goods per year
A Good That
Is Both Inferior
and Normal
Y3 L3
Y2
◦ ….she bought less
hamburger and more
steak.
e3
1
Y1 L
I3
e2
◦ …she bought more
hamburger
e1
I2
I1
Hamburger per year
(b) Engel Curve
Y, Income
But as she became
wealthier and her
income rose…
Income-consumption curve
L2
Y3
E3
Y2
E2
Engel curve
Y1
E1
Hamburger
per year
5-27
Solved Problem
Mahdu views Cragmont and Canada Dry ginger ales as perfect
substitutes: He is indifferent as to which one he drinks.
The price of a 12-ounce can of Cragmont, p, is less than the price of a
12-ounce can of Canada Dry, p*.
What does Mahdu’s Engel curve for Cragmont ginger ale look like?
How much does his weekly ginger ale budget have to rise for Mahdu to
buy one more can of Cragmont ginger ale per week?
5-28
Solved
Problem
5-29
Income and
substitution
effects
30
Income and Substitution
Effects
A change in the price of a good has two effects:
◦ Substitution Effect
◦ Income Effect
31
Income and Substitution
Effects
Substitution Effect
◦ Relative price of a good (PF/PC) changes when price
changes (for example, PF).
32
Income and Substitution
Effects
Income Effect
◦ Consumers experience an increase in real purchasing
power when the price of one good falls
33
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
demanded of the good
34
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 (new budget line)
◦ When a person’s income increases, the quantity
demanded for the product may increase or decrease
depending whether it’s a normal or inferior good.
35
Income and Substitution
Effects
•Income Effect
–For a normal good, the income effect is positive. In
other words, an increase in income causes an
increase in demand, and a decrease in income causes
a decrease in demand.
–For an inferior good, the income effect is negative. In
other words, an increase in income causes a decrease
in demand, and a decrease in income causes an
increase in demand.
36
5-37
5-38
5-39
5-40
D, Movie DVDs, Units per year
Substitution and Income Effects with
Normal Goods
Initial Values
P2 = price of DVDs = $20
P1 = price of CDs = $15
Y = Income = $300.
Budget Line, L
15
L1
P1
q1
q2 =
P2
$300 - $15 q
q2 =
$20 1
$20
Y
P2
e1
-
I1
12
20
C, Music CDs Units peryear
5-41
ovie DVDs, Units per year
Substitution and Income Effects with
Normal Goods
Initial Values
D,
P2 = price of DVDs = $20
P1 = price of CDs = $15
Y = Income = $300.
Budget Line, L
15
L1
Y
P2
q2 =
L2
e2
e1
-
P1
q1
P2
$300 - $15
$30 q
q2 =
$20 1
$20
I1
I2
6
12
Total effect = -6
20
Cq1, Music CDs Units per
year
P1 goes up…
5-42
Substitution and Income Effects with Normal Goods
Initial Values
PD = price of DVDs =
$20
PC = price of CDs = $15
Y = Income = $300.
D, Movie DVDs, Units per year
What if we compensated
Laura so she could afford the
same utility she had before
the price of CDs increased?
L*
◦ In other words, how much
income she would need to
afford indifference curve I1,
with the new price of CDs
($30)
Budget Line, L
PC
C
D=
PD
$300 - $30 C
D=
$20
$20
Y
PD
15
L1
L2
e*
-
e1
e2
I1
I2
6
Income effect = -3
9
12
20
C, Music CDs Units peryear
Substitution effect = -3
Total effect = -6 = Substitution Effect + Income Effect = -3 + (-3)
5-43
Income and Substitution
Effects: Inferior Good
Clothing
(units per
R
month)
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
F1Total Effect
U1
EIncome
S Effect
F2
T
Food (units
per month)
44
Income and Substitution
Effects
A Special Case: The Giffen Good
◦ a special type of inferior good
◦ The negative income effect may theoretically be large
enough to cause the demand curve for a good to slope
upward to overpower the positive substitution effect so
that a fall in price causes an decrease in quantity
demanded, in other words cause the demand curve for
a good to slope upward!
45
Basketball, Tickets per year
Giffen Good
When the price of movie tickets
decreases the budget constraint
rotates out…
L2
e2
L1
I2
e1
Total effect
allowing the consumer to
increase her utility.
I1
Movies, Tickets per year
Nevertheless, the total effect is
negative. WHY?
5-46
Even though the substitution
effect is positive….
Basketball, Tickets per year
Giffen Good
L2
◦ …the income effect is larger and
negative (since this is an inferior
good).
e2
L1
I2
L*
e1
e*
Total effect
Substitution effect
I1
Movies, Tickets per year
Income effect
5-47
Homer Simpson, our representative consumer, consumes varying amounts of beer and pork
rinds. Assume that B = quantity of beer consumed, and that R = quantity of pork rinds
consumed. Homer’s utility function is given as:
Assume further that the price of beer is $4, the price of pork rinds is $2, and that Homer’s
income is $200.
a)Find the optimal bundle for Homer.
Now assume that the price of Pork increases from $2 to $4
b)Find Homer’s new optimal bundle
c)What is the income and the substitution effect in part (b)
d)What if the price of pork rinds goes up, but the government offers to compensate Homer
for this loss of purchasing power. That is, Mayor Quimby offers to mail Homer a check, in an
effort to keep Homer from feeling worse off. Homer still faces the higher pork rinds price,
but doesn’t experience a change in utility. That is, for Homer to be no worse off after the
price increase, the government check must be large enough to keep Homer on his original
indifference curve as in part (a). By how much should the government compensate him?
5-48
Income and Substitution Effects
We have already discussed that a price change has two
effects: (If the price of good 1 falls)
•Substitution Effect: Consumers will tend to buy more of the
good that has become cheaper (good 1), and less of the
good that has become relatively more expensive (good 2).
•Income Effect: Because good 1 is cheaper, consumers enjoy
an increase in real purchasing power.
The trade-off between goods 1 and 2 has changed; another
way to say this is that the relative prices have changed.
5-49
Income and Substitution Effects
(numerical example)
•Recall is the opportunity cost of good 1 in terms of
good 2.
•With old (original) prices p1=p2=$2, then you have to
give up 1 unit of good 2 to get one more unit of good 1.
(Same for good 2)
•With the new prices p1=$1, then ; you only have to give
up ½ unit of good 2 to get an additional unit of good 1
(So good 1 is now relatively cheaper). On the other
hand so that you have to give up 2 units of good 1 to
get an additional unit of good 2. Good 2 is relatively
more expensive now, even though p2 has not changed
5-50
Income and Substitution Effects
(numerical example)
Good 2
m/p2
U’
U
x1
m/p1
x’1
m/p’1 Good 1
5-51
Income and Substitution Effects (numerical
example)
•Originally, m=$20, p1= $2, p2=$2. If Cobb-Douglass
preferences such that U=X1 ½ X2 ½, then from util max, we have
X1*= 0.5m/p1 and X2*= 0.5m/p2 , so optimal bundle is X1*=5,
X2*=5, bundle costs $20, and utility achieved =5½ 5½ =5.
•If p1 falls so p1’=$1, then original bundle now costs only (5)
($1) + (5) ($2) = $15
•Fall in p1 has increased purchasing power (has $5 “extra” to
potentially buy more of each good
•Trade-off between good 1 and good 2 has changed because
p1/p2 changed.
5-52
Income and Substitution
Effects (numerical example)
Back to numerical
example: Now,
p1=$1, p2=$2, m=$20
U(x1, x2) = x1 ½ x2 ½
X2
5
5
10
X1
then from util max,
we have X1*=
0.5m/p1 and X2*=
0.5m/p2 ,
x1*=10,
x2*=5
5-53
Income and Substitution
Effects (numerical example)
Numerically, how do we calculate the changes in demand
due to the substitution effect and the income effect?
When p1 falls from $2 to $1:
The change in demand due to the substitution effect:
p1 falls to p1’
p1 = $2
p1’=$1
p2 = $2
5-54
Income and Substitution
Effects (numerical example)
AB measures the change in demand due to the substitution effect.
The slope of the original budget line is
The slope of the intermediate budget is
A
B
U’
For the intermediate budget constraint, we let the ratio of prices change (from 1 to
1/2) and we find the optimal bundle that gives the same utility as the original
bundle.
5-55
Income and Substitution
Effects (numerical example)
•Recall:
•At the original bundle “A”:
•To calculate the new bundle “B” for the substitution effect, we use the
following two conditions:
•where is the utility achieved by the original optimal bundle “A”. This is the
expenditure min problem!!!
5-56
Income and Substitution
Effects (numerical example)
•So the two conditions become
•Solve for 1 and 2:
x1**=
x2**=
(This is bundle “B”)
•The income required to buy “B” (i.e. the income for the
intermediate budget constraint is:
•($1) (7.1) + ($2)(3.54) = $14.1
p1 x1** p2 x2**
5-57
Income and Substitution
Effects (numerical example)
Original bundle A: (x1, x2) = (5, 5)
Substitution effect: Bundle B: (x1, x2) = (7.1, 3.54)
Change in demand due to the substitution effect
5-58
Income and Substitution Effects
(numerical example)
X2
Red: Original BL
Purple:
Intermediate BL
A
C
B
5
7.1
10
X1
To calculate the change in dd due to the income effect:
BC shows the change in the dd for x1 due to the income effect. Going
from the intermediate budget ($14) to the final budget ($20) represents
an increase in “income” Δ m > 0
5-59
Income and Substitution
Effects (numerical example)
The change in demand due to the income effect:
BC
Bundle B: (x1, x2) = (7.1, 3.54)
C: (x1, x2) = (10, 5)
Δx1n = 10-7.1 = 2.9
All together:
n
Δx1
+
Δx
=
1
Total
change
due to p1
change
Change in x1
due to the
substitution
effect
Change in x1
due to the
income effect
5-60
Cost of Living
Adjustments
61
Inflation Indexes
How accurately government measures inflation?
Inflation - the increase in the overall price level over time.
◦ nominal price - the actual price of a good.
◦ real price - the price adjusted for inflation.
How do we adjust for inflation to calculate the real price?
5-62
Inflation Indexes (cont.)
Consumer Price Index (CPI) – measure the cost of a
standard bundle of goods for use in comparing prices over
time.
◦ We can use the CPI to calculate the real price of a hamburger over
time.
◦ In terms of 2008 dollars, the real price of a hamburger in 1955 was:
CPI for 2008
211.1
 price of a burger 
15  1.18
CPI for 1955
26.8
5-63
Cost-of-Living Indexes
Long term contracts and government programs include COLAs
which raise incomes in proportion to an index of inflation (e.g.
salaries, pensions, social security payments etc)
The CPI is calculated each year as the ratio of the cost of a typical
bundle of consumer goods and services today in comparison to
the cost during a base period.
What you do in Intro Econ to calculate the CPI is: The amount
of money at current year prices that an individual requires to
purchase the bundle of goods and services that was chosen in
the base year divided by the cost of purchasing the same
bundle at base year prices.
Slide 64
Let’s think
Suppose in 2010, income is $80, food costs $5/unit and clothing costs
$5/unit.
If preferences are represented by the Cobb-Douglass utility function
u(F,C) =F1/2C1/2. Then in 2010, the optimal bundle is F=8 and C=8.
In 2015, food costs $6/unit and clothing costs $24/unit.
a. Calculate the CPI (the way you do in Principles of Econ).
b. If income is adjusted according to the CPI, what is the new income?
What is the new optimal bundle?
c. Does the CPI “do the job” correctly, i.e. does it make people as well
off as they were before the price change?
5-65
Cost-of-Living Indexes
What Do You Think?
◦ Does the CPI accurately reflect the cost of living for
retirees?
◦ Is it appropriate to use the CPI as a cost-of-living index
for other government programs, for private union
pensions, and for other private wage agreements?
Slide 66
Cost-of-Living Indexes
Example
◦ Two sisters, Rachel and Sarah, have identical
preferences.
◦ Sarah began college in 1990 with a $500 discretionary
budget.
◦ In 2000, Rachel started college and her parents
promised her a budget that was equivalent in
purchasing power.
Slide 67
Cost-of-Living Indexes
1990 (Sarah) 2000 (Rachel)
Price of books
$20/book
$100/book
Number of books
15
?
Price of food
$2.00/lb.
$2.20/lb
Pounds of food
100
?
Expenditure
$500
?
Slide 68
How much do we compensate
Rachel?
•Price indexes, like the CPI, use a fixed consumption bundle in the base
period.
–
Called a Laspeyres price index
•Suppose we give Rachel enough money in 2000 to buy the same bundle
Sarah bought in 1990. That would be:
$1,720 =100 x 2.20 + 15 x $100
But will Rachel buy the same quantities of goods as her older sister?
5-69
Cost-of-Living Indexes
Books
(per quarter)
Using the Laspeyres
index results in the
budget line shifting
up from I2 to I3.
U1
25
20
A
15
C
10
Sarah bought bundle
A. If the parents gave
enough money to buy
the same bundle,
Rachel would buy
bundle C
L3
5
L2
L1
0 50 100 200 250 300 350 400 450 500 550 600
Slide 70
Food
(lb./quarter)
•The budget with the Laspeyres income adjustment passes through bundle A
but has a different slope (the same bundle is affordable at both sets of prices).
•But Rachel won’t buy the same bundle as her sister. She would buy bundle C
(to equate MRS=MRT).
•This puts Rachel on a higher indifference curve than her older sister.
•The Laspeyres income adjustment OVERCOMPENSATES because it gives higher
utility in the current period than base period.
5-71
Cost-of-Living Indexes
The ideal cost of living index represents the cost of attaining a
given level of utility at current (2000) prices relative to the
cost of attaining the same utility at base (1990) prices.
Slide 72
Cost-of-Living Indexes
Books
(per quarter)
25
For Rachel to achieve
the same level of utility as
Sarah, with the higher
prices, her budget must
be sufficient to allow her
to consume the bundle
shown by point B.
U1
20
A
15
10
B
5
L2
L1
0 50 100 200 250 300 350 400 450 500 550 600
Slide 73
Food
(lb./quarter)
IDEAL Cost-of-Living Indexes
Price of books
1990 (Sarah)
Bundle A
$20/book
2000 (Rachel)
Bundle B
$100/book
Number of books
15
6
Price of food
$2.00/lb.
$2.20/lb
Pounds of food
100
300
Expenditure
$500
$1,260
Slide 74
Cost-of-Living Indexes
Sarah’ Expenditure (at bundle “A”)
$500 = 100 lbs. of food x $2.00/lb. + 15 books x $20/book
Rachel’ Expenditure for Equal Utility (at “B”)
$1,260 = 300 lbs. of food x $2.20/lb. + 6 books x $100/book
Slide 75
Cost-of-Living Indexes
The ideal cost-of-living adjustment for Rachel is $760.
The ideal cost-of-living index is $1,260/$500 = 2.52 or 252.
This implies a 152% increase in the cost of living.
Slide 76
Cost-of-Living Indexes
To do this on an economy-wide basis would entail large amounts of
information.
Price indexes, like the CPI, use a fixed consumption bundle in the base
period.
◦
Called a Laspeyres price index
Slide 77
Cost-of-Living Indexes
Laspeyres Index
The Laspeyres index tells us:
◦ The amount of money at current year prices that an
individual requires to purchase the bundle of goods and
services that was chosen in the base year divided by the
cost of purchasing the same bundle at base year prices.
Slide 78
Cost-of-Living Indexes
Calculating Rachel’s Laspeyres cost of living index
◦ Setting the quantities of goods in 2000 equal to what
were bought by her sister, but setting their prices at
their 2000 levels result in an expenditure of $1,720
(100 x 2.20 + 15 x $100)
Slide 79
Cost-of-Living Indexes
Her cost of living adjustment would now be $1,220.
The Laspeyres index is: $1,720/$500 = 344.
This overstates the true cost-of-living increase.
Slide 80
Cost-of-Living Indexes
What Do You Think?
◦ Does the Laspeyres index always overstate the true cost-ofliving index?
Slide 81
Cost-of-Living Indexes
Yes!
◦ The Laspeyres index assumes that consumers do not alter
their consumption patterns as prices change.
Slide 82
Cost-of-Living Indexes
Yes!
◦ By increasing purchases of those items that have become
relatively cheaper, and decreasing purchases of the
relatively more expensive items consumers can achieve the
same level of utility without having to consume the same
bundle of goods.
Slide 83
Cost-of-Living Indexes
The Paasche Index
◦ Calculates the amount of money at current-year prices that
an individual requires to purchase a current bundle of goods
and services divided by the cost of purchasing the same
bundle in the base year.
Slide 84
Cost-of-Living Indexes
Comparing the Two Indexes
•Both indexes involve ratios that involve today’s current year
prices, PFt and PCt and base year prices PFb and PCb
•However, the Laspeyres index relies on base year
consumption, Fb and Cb.
•Whereas, the Paasche index relies on today’s current
consumption, Ft and Ct .
Slide 85
Cost-of-Living Indexes
Comparing the Two Indexes
Let:
◦ PFt & PCt be current year prices
◦ PFb & PCb be base year prices
◦ Ft & Ct be current year quantities
◦ Fb & Cb be base year quantities
Slide 86
Cost-of-Living Indexes
Then a comparison of the Laspeyres and Paasche indexes
gives the following equations:
Slide 87
Cost-of-Living Indexes
Comparing the Two Indexes
Sarah (1990)
Slide 88
Cost-of-Living Indexes
The Paasche Index
Sarah (1990)
◦ Cost of buying current year bundle at current year prices is
$1,260 (300 lbs x $2.20/lb + 6 books x $100/book)
◦ Cost of the same bundle at base year prices is $720 (300 lbs
x $2/lb + 6 books x $20/book)
Slide 89
Cost-of-Living Indexes
Sarah (1990)
Comparing the Two Indexes
Slide 90
Cost-of-Living Indexes
The Paasche Index
The Paasche index will understate the cost of living
because it assumes that the individual will buy the current
year bundle in the base year.
Slide 91
Deriving Labor Supply Curves
Previously, we used concepts like indifference curves, budget
constraints and consumer equilibria to study the individual’s
demand for goods
We can also apply the consumer theory model to derive the
supply curve of labor
As the name implies, the two goods we will look at are
consumption and leisure. We will continue to make the same
preference assumptions we used before.
We do so by using consumer theory to obtain a demand
curve for leisure time and then using the demand curve to
derive the supply curve of hours spent working
Deriving Labor Supply Curves
People choose between working to earn money to buy
goods and services and consuming leisure: all their time
spent not working for pay
Jackie spends her total income, Y on various goods. For
simplicity assume that the price of these goods is $1 per
unit, so she buys Y goods
Her utility, U, depends on how many goods, Y, and how
much leisure, N, he consumes
Labor-Leisure Choice
Income can vary depending on how much the individual works. Instead,
the individual is constrained by her time endowment, T, usually 24
hours (but we can make it less).
Leisure - all time spent not working.
Labor plus Leisure is always equal to the time endowment.
The number of hours worked per day, H, equals 24 minus the hours of
leisure or nonwork, N, in a day:
H = 24 − N.
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Labor-Leisure Choice: Example
Jackie spends her total income, Y, on various goods.
◦ The price of these goods is $1 per unit.
Her utility, U, depends on how many goods and how much leisure
she consumes:
U = U(Y, N).
Jackie’s earned income (his wage times the number of hours he
works) equal:
wH.
And her total income, Y, is her earned income plus her unearned
income, Y* (such as from an inheritance or a gift from his parents):
Y = wH + Y*.
5-95
Labor-Leisure Choice: Example
Using consumer theory, we can determine Jackie’s
demand curve for leisure once we know the price of
leisure
The wage, w, is the cost: what you would have earned
from an hour’s work: the price of leisure is forgone
earnings
The higher the wage, the more an hour of leisure costs
you
The wage rate, w, is measured in $/hour.
Y, Goods per day
(a) Indifference Curves and Constraints
Demand for
Leisure
Time constraint
I1
Budget Line, L1
L1
–w1
Y = w1H
Each extra
hour of leisure
she consumes
costs her w1
goods.
Y1
0
24
e1
N1 = 16
H1 = 8
24
0
N, Leisure hours per day
H, Work hours per day
(b) Demand Curve
w, Wage per hour
Y = w1(24 − N).
1
E1
w1
0
N1 = 16
H1 = 8
N, Leisure hours per day
H, Work hours per day
5-97
Labor-Leisure Choice: Example
We can solve this problem using calculus
Jackie maximizes her utility subject to the time
constraint and the income constraint
By using the chain rule of differentiation, we find
that the first-order condition for an interior
maximum to the problem is:
Y, Goods per day
(a) Indifference Curves and Constraints
Demand for Leisure
L2
–w2
I1
1
e2
Y2
Budget Line, L1
Y = w1 H
Y = w1(24 − N).
L1
–w1
1
w 2 > w1
N2 = 12
H2 = 12
N1 = 16
H1 = 8
24
0
N, Leisure hours per day
H, Work hours per day
(b) Demand Curve
w, Wage per hour
Y = w2 H
Y = w2(24 − N).
e1
Y1
0
24
Budget Line, L2
Time constraint
I2
E2
w2
E1
w1
Demand for leisure
0
N2 = 12
H2 = 12
N1 = 16
H1 = 8
N, Leisure hours per day
H, Work hours per day
5-99
Labor-Leisure Choice: Example
By subtracting Jackie’s demand for leisure at each
wage - her demand curve for leisure - we construct
her labor supply curve the hours she is willing to
work as a function of the wage, H(w)
Her supply curve for hours worked is the mirror
image of the demand curve for leisure: For every
extra hour of leisure that Jackie consumes, she
works one hour less
Supply Curve of Labor
5-101
Y, Goods per d ay
Income and Substitution Effects of
a Wage Change
Time const raint
I2
L2
Since income effect is
positive, leisure is a
normal good.
I1
L*
e2
e*
L1
e1
0
N*
24
H*
N1 N 2
H1 H2
24
N, Leisure hours per d ay
0
H, Work hours per d ay
Substitution effect Total effect
Income effect
5-102
Solved Problem
Enrico receives a no-strings-attached scholarship that pays him an extra
Y* per day. How does this scholarship affect the number of hours he
wants to work? Does his utility increase?
5-103
Solved Problem
5-104
Income and Substitution
Effects of a Wage Change
We’ve seen that when the wage rate changes, leisure can
increase or decrease depending on whether the income or
substitution effects dominate.
It could also be possible that at some wage rates the income
effect dominates and at other wage rates the substitution effect
dominates.
It seems plausible that at low wage rates the substitution effect
would dominate (upward sloping labor supply) and at high wage
rates the income effect would dominate (downward sloping labor
supply).
If this were the case, the individuals labor supply curve would be
backward bending.
Labor Supply Curve That Slopes Upward and Then
Bends Backward
L3
(b) Supply Curve of Labor
Time const raint
I3
I2
I1
L2
E3
E2
e3
e2
E1
L1
24
Supply curve of labor
w, Wage per hour
Y, Goods per d ay
(a) Labor-Leisure Choice
e1
H2
H
3
H1
0
H, Work hours per d ay
0
H1 H3
H2
24
H, Work hours per d ay
butlow
At
at high
wages,
wages,
an increase
an increase
in the wage causes the worker
worker
to
work to
less….
work more….
5-106