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SLIDES BY
John Loucks
St. Edward’s
University
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 1
Chapter 20
Index Numbers

Price Relatives

Aggregate Price Indexes

Computing an Aggregate Price Index
from Price Relatives

Some Important Price Indexes

Deflating a Series by Price Indexes

Price Indexes: Other Considerations

Quantity Indexes
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 2
Price Relatives

Price relatives are helpful in understanding and
interpreting changing economic and business
conditions over time.

A price relative shows how the current price per unit
for a given item compares to a base period price per
unit for the same item.
A price relative expresses the unit price in each period
as a percentage of the unit price in the base period.


A base period is a given starting point in time.
Price in period t
Price relative in period t =
( 100)
Base period price
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 3
Price Relatives

Example: Besco Products
The prices Besco paid for newspaper and
television ads in 2001 and 2011 are shown below.
Using 2001 as the base year, compute a 2011 price
index for newspaper and television ad prices.
Newspaper
Television
2001
2011
$14,794
11,469
$29,412
23,904
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 4
Price Relatives
Newspaper
I 2011
29, 412

(100)  199
14,794
Television
I 2011
23,904

(100)  208
11, 469
Television advertising cost increased at a greater rate.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 5
Aggregate Price Indexes

An aggregate price index is developed for the specific
purpose of measuring the combined change of a
group of items.

An unweighted aggregate price index in period t,
denoted by It , is given by
 Pit
It 
( 100)
 Pi 0
where
Pit = unit price for item i in period t
Pi 0 = unit price for item i in the base period
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 6
Aggregate Price Indexes

With a weighted aggregate index each item in the
group is weighted according to its importance, which
typically is the quantity of usage.

Letting Qi = quantity for item i, the weighted aggregate
price index in period t is given by
 Pit Q i
It 
( 100)
 Pi 0 Q i
where the sums are over all items in the group
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 7
Aggregate Price Indexes

When the fixed quantity weights are determined
from the base-year usage, the index is called a
Laspeyres index.

When the weights are based on period t usage, the
index is a Paasche index.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 8
Aggregate Price Indexes

Example: City of Rockdale
Data on energy consumption and expenditures
by sector for the city of Rockdale are given on the
next slide. Construct an aggregate price index for
energy expenditures in 2011 using 1990 as the base
year.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 9
Aggregate Price Indexes

Example: City of Rockdale
Sector
Residential
Commercial
Industrial
Transport.
Quantity (BTU) Unit Price ($/BTU)
1990
2011
1990
2011
9,473
8,804
$2.12
$10.92
5,416
6,015
1.97
11.32
21,287 17,832
.79
5.13
15,293 20,262
2.32
6.16
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 10
Aggregate Price Indexes

Example: City of Rockdale
Unweighted Aggregate Price Index
10.92  11.32  5.13  6.16
I 2011 
(100)  466
2.12  1.97  .79  2.32
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 11
Aggregate Price Indexes

Example: City of Rockdale
Weighted Aggregate Index (Laspeyres Method)
10.92(9, 473)   6.16(15, 293)
I 2011 
(100)  443
2.12(9, 473)   2.32(15, 293)
Weighted Aggregate Index (Paasche Method)
10.92(8,804)   6.16(20, 262)
I 2011 
(100)  415
2.12(8,804)   2.32(20, 262)
The Paasche value being less than the
Laspeyres indicates usage has increased
faster in the lower-priced sectors.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 12
Aggregate Price Indexes
Example: Annual Cost of Lawn Care
Dina Evers is pleased with her lovely lawn, but she
is concern about the increasing cost of maintaining it.
The cost includes mowing, fertilizing, watering, and
more.
Dina wants an index that measures the change in
the overall cost of her lawn care. Price and quantity
data for her annual lawn expenses are listed on the
next slide.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 13
Aggregate Price Indexes
Example: Annual Cost of Lawn Care
Quantity
Item
(Units)
Mowing
32
Leaf Removal
3
Watering (1000s gal.)
40
Fertilizing
2
Sprinkler Repair
1
Unit Price ($)
2007
2011
57.00
79.00
56.00
71.00
1.83
2.78
56.00
67.00
109.00 128.00
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 14
Aggregate Price Indexes
Unweighted
Unweighted aggregate price index in period t is:
It
P


P
it
(100)
i0
where:
Pit = unit price for item i in period t
Pi0 = unit price for item i in the base period
A 24%
increase
in annual
lawn care
expenses
79.00  71.00  2.78  67.00  128.00
I 2011 
(100)  124
57.00  56.00  1.83  56.00  109.00
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 15
Aggregate Price Indexes
Weighted (Fixed Quantity)
Weighted aggregate price index in period t is:
It
PQ


(100)
P Q
it
i
i0
i
where: Qi = quantity for item i
I 2011 
Note that Qi
does not have a
second subscript
for a time period.
79.00(32)  71.00(3)  2.78( 40)  67.00(2)  128.00(1)
(100)  136
57.00(32)  56.00(3)  1.83( 40)  56.00(2)  109.00(1)
A 36% increase in
annual lawn care expenses
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 16
Aggregate Price Indexes
Weighted (Base-Period Quantity)
Special case
of the fixed
quantity index
Pit Qi 0
Laspeyres

It 
(100)
Index
 Pi 0Qi 0
More widely
used than the
Paasche index
Weighted (Period t Quantity)
A variablequantity index
Paasche
Index
It
PQ


P Q
it
it
i0
it
(100)
Pro: Reflects
current usage;
Con: Weights
require continual updating
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 17
Aggregate Price Indexes
Item i
Mowing
Leaf Removal
Water (1000s gal.)
Fertilizing
Sprinkler Repair
Unit Price ($)
2007
2011
Pi0
Pit
57.00
56.00
1.83
56.00
109.00
79.00
71.00
2.78
67.00
128.00
Price
Relative
(Pit/Pi0)100
138.6
126.8
151.9
119.6
117.4
79
(100)
57
The 5-year increases in unit price ranged from a low of
17.4% for sprinkler repair to a high of 51.9% for water.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 18
Aggregate Price Indexes
Item
Mowing
Leaves
Water
Fertilize
Sprinkler
Price
Base
Weighted
Relative Price ($) Quantity Weight Price Relative
Pi0
Qi
wi = Pi0Qi (Pit/Pi0)(100)wi
(Pit/Pi0)100
138.6
126.8
151.9
119.6
117.4
57.00
56.00
1.83
56.00
109.00
32
3
40
2
1
1,824.0
168.0
73.2
112.0
109.0
252,806.40
21,302.40
11,119.08
13,395.20
12,796.60
Pit
Total 2,286.2
311,419.68
(
100
)
w
P
i
311, 419.68
i0
This value is the same as the
It 

 136
2, 286.2
one identified by the weighted
 wi
aggregate index computation.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 19
Some Important Price Indexes

Consumer Price Index (CPI)
• Primary measure of the cost of living in U.S.
•
Based on 400 items including food, housing,
clothing, transportation, and medical items.
•
Weighted aggregate price index with fixed weights
derived from a usage survey.
•
Published monthly by the U.S. Bureau of Labor
Statistics.
•
Its base period is 1982-1984 with an index of 100.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 20
Some Important Price Indexes
Consumer Price Index (CPI)
Base 1982-1984 = 100.0
Year
1980
1981
1982
1983
1984
1985
1986
1987
CPI
82.4
90.9
96.5
99.6
103.9
107.6
109.6
113.6
Year
1988
1989
1990
1991
1992
1993
1994
1995
CPI
118.3
124.0
130.7
136.2
140.3
144.5
148.2
152.4
Year
1996
1997
1998
1999
2000
2001
2002
2003
CPI
156.9
160.5
163.0
166.6
172.2
177.1
179.9
184.0
Year
2004
2005
2006
2007
2008
2009
2010
2011
CPI
188.9
195.3
201.6
207.3
215.3
214.5
218.1
224.9
Note: For 1982 – 1984, (96.5 + 99.6 + 103.9)/3 = 100.0
Also note: CPI for 2009 was lower than CPI for 2008.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 21
Some Important Price Indexes

Producer Price Index (PPI)
• Measures the monthly changes in prices in primary
markets in the U.S.
• Used as a leading indicator of the future trend of
consumer prices and the cost of living.
• Covers raw, manufactured, and processed goods at
each level of processing.
•
•
Includes the output of manufacturing, agriculture,
forestry, fishing, mining, gas and electricity, and
public utilities.
Is a weighted average of price relatives using the
Laspeyres method.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 22
Some Important Price Indexes

Dow Jones Averages
• Indexes designed to show price trends and
movements on the New York Stock Exchange.
• The Dow Jones Industrial Average (DJIA) is based
on common stock prices of 30 industrial firms.
• The DJIA is not expressed as a percentage of baseyear prices.
• Another average is computed for 20 transportation
stocks, and another for 15 utility stocks.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 23
Some Important Price Indexes
Dow Jones Industrial Average (DJIA)
30 Companies
3M
Alcoa
DuPont
ExxonMobil
McDonald's
American Express
General Electric
Microsoft
At&T
Hewlett-Packard
Pfizer
Bank of America
Home Depot
Procter & Gamble
Boeing
IBM
Travelers
Caterpillar
Intel
United Technologies
Chevron Corp.
Johnson & Johnson
Verizon
Cisco Systems
J.P.Morgan Chase
Wal-Mart Stores
Coca-Cola
Kraft Foods
Walt Disney
Merck
As of 09/2012
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 24
Deflating a Series by Price Indexes

In order to correctly interpret business activity over
time when it is expressed in dollar amounts, we
should adjust the data for the price-increase effect.

Removing the price-increase effect from a time series
is called deflating the series.

Deflating actual hourly wages results in real wages
or the purchasing power of wages.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 25
Deflating a Series by Price Indexes

Example: McNeer Cleaners
McNeer Cleaners, with 46 branch locations, has
had the total sales revenues shown on the next
slide for the last five years. Deflate the sales revenue
figures on the basis of 1982-1984 constant dollars. Is
the increase in sales due entirely to the price-increase
effect?
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 26
Deflating a Series by Price Indexes

Example: McNeer Cleaners
Year
Total Sales ($1000)
CPI
2007
2008
2009
2010
8,446
9,062
9,830
10,724
207.3
215.3
214.5
218.1
2011
11,690
224.9
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 27
Deflating a Series by Price Indexes

Adjusting Revenue For the Price-Increase Effect
Year
Deflated
Sales ($1000)
Annual
Change (%)
2007
2008
2009
2010
2011
(8,446/207.3)(100) = 4,074
(9,062/215.3)(100) = 4,209
(9,830/214.5)(100) = 4,583
(10,724/218.1)(100) = 4,917
(11,690/224.9)(100) = 5,198
+3.3
+8.9
+7.3
+5.7
After adjusting, revenue is still increasing at an
average rate of 6.3% per year.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 28
Real Sales Revenue ($1000s)
Deflating a Series by Price Indexes
5500
5198
5250
5000
4583
4750
4500
A real sales
increase of
27.6% from
2007 to 2011
4917
4209
4074
4250
2007
2008
2009
Year
2010
2011
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 29
Price Indexes: Other Considerations
Selection of Items
When the class of items is very large, a
representative group (usually not a random sample)
must be used.
The group of items in the aggregate index must be
periodically reviewed and revised if it is not
representative of the class of items in mind.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 30
Price Indexes: Other Considerations
Selection of a Base Period
As a rule, the base period should not be too far from
the current period.
For example, a CPI with a 1945 base period would
be difficult for most individuals to understand, due
to unfamiliarity with conditions in 1945.
The CPI’s
base period
The base period for most indexes is
was changed
adjusted periodically to a more recent
to 1982-84
period of time.
in 1988.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 31
Price Indexes: Other Considerations
Quality Changes
A basic assumption of a price index is that prices
over time are identified for the same item.
Is a product that has undergone a major quality
change the same product it was?
An increase in an item’s quality may or may not
result in a price increase . . . and a decrease in
quality may or may not result in a price decrease.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 32
Quantity Indexes
An index used to measure changes in quantity levels
over time is called a quantity index.
A quantity relative shows how the current quantity
level for a single item compares to a base period
quantity level for the same item.
A weighted aggregate quantity index is computed
in much the same way as a weighted aggregate
price index.
 Q it w i
It 
(100)
 Qi0 w i
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 33
Quantity Indexes
Example: Appliance Mart
Appliance Mart reports the 2002 and 2011 sales for
three major kitchen appliances as shown below.
Compute quantity relatives and use them to develop
a weighted aggregate quantity index for 2011.
Kitchen
Appliance
Dishwasher
Range
Refrigerator
Sales Price Sales (Units)
2002
2011
(2002)
$325
$450
$710
720
540
980
950
610
1110
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 34
Quantity Indexes
Quantity in period t
Quantity relative in period t =
(100)
Base period quantity
Kitchen
Appliance
Sales (Units)
2002
2011
Dishwasher
Range
Refrigerator
720
540
980
950
610
1110
Quantity Relative
(950/720)100 = 131.9
(610/540)100 = 113.0
(1110/980)100 = 113.3
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 35
Quantity Indexes
Unweighted
Unweighted aggregate quantity index in period t is:
It
Q


Q
it
(100)
i0
where: Qit = quantity for item i in period t
Qi0 = quantity for item i in the base period
I 2011 
950  610  1110
(100)  119
720  540  980
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 36
Quantity Indexes
Weighted (Base-Period Price)
Weighted aggregate quantity index in period t is:
It
QP


Q P
it i 0
(100)
i0 i0
where: Pi 0 = price for item i in the base period
950(325)  610( 450)  1110(710)
I 2011 
(100)  117
720(325)  540( 450)  980(710)
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 37
Quantity Indexes
Weighted (Fixed Price)
It
Q P


(100)
Q P
it i
i0 i
Note that Pi
does not have a
second subscript
for a time period.
Weighted (Period t Price)
It
Q P


Q P
it it
(100)
i 0 it
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 38
End of Chapter 17
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 39