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```Index Numbers
Chapter 17
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Learning Objectives
LO17-1
LO17-2
LO17-3
LO17-4
LO17-5
17-2
Compute and interpret a simple, unweighted
index
Compute and interpret an unweighted aggregate
index
Compute and interpret a weighted aggregate
index
List and describe special-purpose indexes
Apply the Consumer Price Index
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Index Numbers
INDEX NUMBER A number that expresses the relative change in price,
quantity, or value compared to a base period.
The major characteristics of an index are
 It is a percentage, but the percent sign is usually
omitted
 It has a base period
 The reasons for computing an index are
 To facilitate the comparison of unlike series
 If the numbers are very large, often it is easier to
comprehend the change of the index rather than the
numbers
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
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Hourly Wages Index Example
According to the Bureau of Labor Statistics, in 2000, the average hourly earnings of
production workers was \$14.02. In January 2019, it was \$27.56. What is the index of
hourly earnings of production workers for January 2019 based on 2000 data?
P = (100) = (100) = 196.58
Thus, the hourly earnings in 2019 compared to 2000 were 196.58%. This means
there was a 96.58% increase in hourly earnings during the period, found by
196.58 – 100.0 = 96.58
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Population Index Example
An index can also compare one item to another. The population of the Canadian
province of British Columbia in 2019 was 4,862,610, and for Ontario it was
14,374,084. What is the population index of British Columbia compared to Ontario?
The index of population for British Columbia is 33.9 found by:
P = (100) = (100) = 33.8
The population of British Colombia is 33.8% (about one-third) of the population of
Ontario, or another way to say that is the population of British Columbia is 66.2%
less than the population of Ontario (100 – 33.8 = 66.2)
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Airport Index Example
The following chart shows the number of passengers (in millions) for the 10 busiest
airports in the United States in 2014. Use the McCarran International Airport in
Las Vegas as the base. What is the index for the other airports compared to
Las Vegas?
To find the 10 indexes, divide the passengers for Las Vegas into the
passengers for the other 9 airports. So the index for Atlanta is 224.2
found by (96.2 &divide; 42.9) &times; (100) = 224.2
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E-Commerce Sales Index Example

Converting data to indexes makes it easier to see the
trend in a series of very large numbers
For example
 US retail e-commerce sales in 2018 were \$504,582,000
 In 2010, e-commerce sales were \$168,895,000
 An increase of \$335,687,000
Index =
2018 e−commerce sales
\$504,582,000
(100) =
(100) = 298.8
2010 e−commerce sales
\$168,895,000
This means 2018 sales increased 198.8% in the eight-year period.
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Construction of Index Numbers

To calculate a simple price index P, using 100 as the base
value for any given year, use the formula below
Suppose the price of a fall weekend package at Tryon Mountain Lodge in western
North Carolina in 2000 was \$450. The price rose to \$795 in 2019. What is the price
index for 2019 using 2000 as the base period and 100 as the base value?
P
\$795
P = P t (100) =
(100) = 176.7
\$450
0
The fall weekend package increased 76.7% from 2000 to 2019.
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Converting to Indexes using Different Base
Periods
Below is a table for prices of a Benson Automatic Stapler, Model 3, converted to
indexes using three different base periods. First, a single year (2015) is used and
each year’s price is divided by 20.
Next, two years (2015-2016) are used as the base; the base price of the stapler
would be \$21, found by averaging the price of the stapler in the two years (20 +
22) &divide; 2 = \$21and then dividing each year’s price by 21.
Finally, the prices \$20, \$22, \$23 are averaged if we use three years (2015-2017) as
the base and then each year’s price is divided by 21.67 to obtain the price index.
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Unweighted Indexes



In an unweighted index, we do not consider the quantities
We may wish to combine several related items and
compare this group of items in two different time periods
 An index for items related to owning and operating an
automobile (tires, oil changes, and gasoline)
 An index for items related to expenses of a college
student (books, tuition, housing, meals, entertainment)
In the simple average of price indexes, we add the simple
indexes for each item and divide by the number of items
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Simple Average of Price Indexes Example
This table reports the prices for several food items in 2009 and 2019. We
would like to develop an index for this food group for 2019 using 2009 prices
as the base. This is written 2009=100
First, we use formula 17-1 to compute the simple index for each food item.
P
1.274
For instance, the index for bread, P = P t (100) = 1.381 (100) = 92.3
0
Now use formula 17-2 to compute the percentage change in the group.
ΣP 92.3+ …. +147.9 591.0
P= i=
=
= 98.5
n
6
6
The mean price of food decreased 1.5% 2009 to 2019.
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Simple Aggregate Price Index

In a simple aggregate price index, the prices of the items
in the group are totaled for both periods and compared
The simple aggregate index for food items on the previous slide is found by
dividing the sum of prices in 2019 by the sum of the prices in 2009.
P = (100) = (100) = 103.7
This means that the aggregate group of prices had increased 3.7% from 2009 to
2019.
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Weighted Indexes: Laspeyres Method




In a weighted index, the quantities are considered
In the Laspeyres method, base period quantities are used
in both the base period and the given period
 Only quantity data from the base period is used which
allows for more meaningful comparison over time
 Does not reflect changes in buying patterns over time
 It may overweight goods whose prices increase
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Laspeyres Price Index Example
First we determine the total amount spent for the six items in the base period, 2009.
To find this value, multiply the base period (2009) price for bread, \$1.381, by the base
period quantity of 50. The result is \$69.2. Continue that for all items and total the
result. The base period total is \$648.23.
The current year total is computed in a similar fashion. For bread, we multiply the
quantity in 2009 by the price of bread in 2019; 50 times \$1.274 is \$70.07. We make the
same calculation for the other items and total to get \$667.783.
P = (100) = (100) = 103.0
We conclude the price
of this group of items
has increased 3.0% from
2009 to 2019.
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Weighted Indexes: Paasche Method

In the Paasche method, current period quantities are used

 Current buying habits are reflected
 It requires quantity data for the current year and it
tends to overweight goods whose prices have declined
 It requires the product of prices and quantities to be
recomputed each year

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Paasche Price Index Example
The following table shows the calculations to determine the Paasche index.
Σp q
\$667.78
P = t t (100) =
(100) = 88.7
Σp0qt
\$753.00
This result indicates that there has been an decrease of 11.3% in the price of this
“market basket” of goods between 2009 and 2019.
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Fisher’s Ideal Index

Fisher’s ideal index is the geometric mean of the
Laspeyres and Paasche indexes
Determine Fisher’s ideal index for the data in Table 17-3 using our results
from the earlier examples.
Fisher’s ideal index = )
=
So, Fisher’s ideal index is 95.6
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Value Index

A value index uses both base period and current period
prices and quantities
The prices and quantities sold at the Waleska Clothing Emporium for ties, suits,
and shoes for May 2015 and May 2019 are given in the table on the left.
What is the index of value for May 2019 using May 2015 as the base period?
Total sales in May 2019 were \$10,600 and in 2015 is \$9,000.
Σp q
\$10,600
V = Σp tqt (100) =
(100) = 117.8
\$9,000
0 0
The value of apparel sales increased 17.8% from May 2015 to May 2019.
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Special-Purpose Indexes

Here is an example of using an index as a measure of
general business activity for northwest US
The Seattle Chamber of Commerce wants to develop a measure of general business
activity. It will be called the General Business Activity Index of the Northwest and will
(10%), and exports from Seattle harbor (20%).
increased 57.0% from
2005 to 2010 and 57.1%
from 2005 to 2018.
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Consumer Price Index
The BLS reports
this index monthly.
It describes the
changes in prices
from one period to
another for a
goods and services.
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Producer Price Index
The PPI reflects the
prices of over 3,400
commodities and price
data is collected
from the sellers. The PPI
is a Laspeyres type
index.
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Dow Jones Industrial Average (DJIA)
This is an index of stock prices. It represents the prices of 30
specific stocks of large publicly owned U.S. based companies.
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Consumer Price Index (CPI)

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


Prices are collected on about 80,000 items monthly from retail
stores, service establishments, rental units, and doctors offices
The CPI is often used to show the rate of inflation in the
United States
It is reported monthly by the U.S. Bureau of Labor Statistics
The current base period is
1982-84
The CPI is used to compute
and tax brackets
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Real Income
Suppose Ms. Watts earned \$20,000 per year in the base period of 1982, 1983, and
1984. She has a current income of \$40,000. Note that although her money
income has doubled since the base period of 1982-84, the prices she pays for
food, gasoline, clothing, and other items has also doubled. Compute her real
income.
Ms. Watts purchasing power (real income) has remained the same at \$20,000.
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Deflating Sales
The sales of Hill Enterprises, a small injection molding company in upstate New
York, increased from 1982 to 2018. The owner, Harry Hill, realizes that the price
of raw materials used in the production process has also increased, so Mr. Hill
wants to deflate sales to account for the increase in raw materials. What are the
deflated sales for 1990, 2000, 2005, 2010, 2015, and 2018 expressed in constant
1982 dollars?
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Suppose the Consumer Price Index this month is 200.0 (1982-84 = 100).
What is the purchasing power of the dollar?
Purchasing power of dollar = (100) = (100) = \$0.50
The CPI of 200 indicates that prices have doubled from the years 1982-84 to
this month. Thus, the purchasing power of the dollar has been cut in half.
That is, a 1982-84 dollar is only worth 50 cents this month.
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Chapter 17 Practice Problems
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Question 1
LO17-1
PNC Bank Inc., which has its headquarters in Pittsburgh,
reported the following commercial loan totals for years
2010 through 2018. Using 2010 as the base, develop a
simple index of commercial loans for the following years
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Question 5
LO17-1,2,3
Following are the prices of toothpaste (9 oz.), shampoo (7
oz.), cough tablets (package of 100), and antiperspirant (2
oz.) for August 2010 and August 2018. Also included are the
quantity purchased. Use August 2010 as the base.
a.
b.
c.
d.
e.
Determine the simple price indexes.
Determine the simple aggregate price index for the two years.
Determine the Laspeyres price index.
Determine the Paasche price index.
Determine Fisher’s ideal index.
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Question 11
LO17-4
You want to construct a leading indicator for Erie County in upstate New
York. The index is to be based on 2000 data. Because of the time and work
involved, you decide to use only four time series. As an experiment, you select
these four series: unemployment in the county, a composite index of county
stock prices, the County Price Index, and retail sales. Here are the figures for
2000 and 2018
The weights you assign are unemployment rate 20%, stock prices 40%, County
Price Index 25%, and retail sales 15%.
a.
Using 2000 as the base period, construct a leading economic indicator for
2018.
b.
index.
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Question 15
LO17-5
In 2000, the mean salary of classroom teachers in Tinora
School District was \$28,650. By 2010, the mean salary
increased to \$33,972, and further increased in 2018 to
\$37,382. The American Federation of Classroom Teachers
maintains information on the trends throughout the United
States in classroom teacher salaries. Its index, which has a
base period of 2000, was 122.5 for 2010 and 136.9 for 2018.
Compare the Tinora teachers to the national trends.
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