Introduction to
Statistics
Sayeeda Jahan
Spring 2021
Slide 1
Chapter 17
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
Slide 2
Price Relatives
Price relatives are helpful in understanding and
interpreting changing economic and business
conditions over time.
Slide 3
Price Relatives
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
Slide 4
Example: Besco Products
Price Relatives
The prices Besco paid for newspaper and
television ads in 1992 and 1997 are shown below.
Using 1992 as the base year, compute a 1997 price
index for newspaper and television ad prices.
1992 1997
Newspaper $14,794 $29,412
Television 11,469 23,904
Slide 5
Example: Besco Products
Price Relatives
Newspaper Television
29,412
I1997 (100) 199
14,794
Television advertising cost increased at a greater rate.
Slide 6
Aggregate Price Indexes
An aggregate price index is developed for the specific
purpose of measuring the combined change of a
group of items.
An unweighted aggregate price index in period t,
denoted by It , is given by
Pit
It (100)
Pi 0
where
Pit = unit price for item i in period t
Pi 0 = unit price for item i in the base period
Slide 7
Aggregate Price Indexes
With a weighted aggregate index each item in the
group is weighted according to its importance, which
typically is the quantity of usage.
Letting Qi = quantity for item i, the weighted
aggregate price index in period t is given by
Pit Q i
It (100)
Pi 0 Q i
where the sums are over all items in the group.
Slide 8
Aggregate Price Indexes
When the fixed quantity weights are determined
from the base-year usage, the index is called a
Laspeyres index.
When the weights are based on period t usage the
index is a Paasche index.
Slide 9
Example: City of Newton
Aggregate Price Indexes
Data on energy consumption and expenditures
by sector for the city of Newton are given below.
Construct an aggregate price index for energy
expenditures in 2000 using 1985 as the base year.
Quantity (BTU) Unit Price ($/BTU)
Sector 1985 2000 1985 2000
Residential 9,473 8,804 $2.12 $10.92
Commercial 5,416 6,015 1.97 11.32
Industrial 21,287 17,832 .79 5.13
Transport. 15,293 20,262 2.32 6.16
Slide 10
Example: City of Newton
Unweighted Aggregate Price Index
I2000 = 10.92 + 11.32 + 5.13 + 6.16 (100) = 466
2.12 + 1.97 + .79 + 2.32
Weighted Aggregate Index (Laspeyres Method)
I2000 = 10.92(9473) + . . . + 6.16(15293) (100) = 443
2.12(9473) + . . . + 2.32(15293)
Weighted Aggregate Index (Paasche Method)
I2000 = 10.92(8804) + . . . + 6.16(20262) (100) = 415
2.12(8804) + . . . + 2.32(20262)
The Paasche value being less than the Laspeyres
indicates usage has increased faster in the lower-
priced sectors.
Slide 11
Computing an Aggregate Price Index from
Price Relatives
Let,
Wi = the weight applied to the price relative for item
i. The general expression for a weighted average of
price relatives is given by
Where,
Slide 12
Some Important Price Indexes
Consumer Price Index (CPI)
• Primary measure of the cost of living in US.
• 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 US Bureau of Labor
Statistics.
• Its base period is 1982-1984 with an index of 100.
Slide 13
Some Important Price Indexes
Producer Price Index (PPI)
• Measures the monthly changes in prices in
primary markets in the US.
• 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.
• Weighted average of price relatives using the
Laspeyres method.
Slide 14
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 base-
year prices.
• Another average is computed for 20
transportation stocks, and another for 15 utility
stocks.
Slide 15
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.
Slide 16
Example: McNeer Cleaners
Deflating a Series by Price Indexes
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?
Slide 17
Example: McNeer Cleaners
Deflating a Series by Price Indexes
Year Total Sales ($1000) CPI
1996 8,446 156.9
1997 9,062 160.5
1998 9,830 163.0
1999 10,724 166.6
2000 11,690 172.6
Slide 18
Example: McNeer Cleaners
Deflating a Series by Price Indexes
Deflated Annual
Year Sales ($1000) Change(%)
1996 (8,446/156.9)(100) = 5,383
1997 (9,062/160.5)(100) = 5,646 +4.9
1998 (9,830/163.0)(100) = 6,031 +6.8
1999 (10,724/166.6)(100) = 6,437 +6.7
2000 (11,690/172.6)(100) = 6,773 +5.2
After adjusting revenue for the price-increase
effect, revenue is still increasing at an average rate of
5.9% per year.
Slide 19
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.
Selection of a Base Period
• As a rule, the base period should not be too far
from the current period.
• The base period for most indexes is adjusted
periodically to a more recent period of time.
Slide 20
Price Indexes: Other Considerations
Quality Changes
• A basic assumption of price indexes is that the
prices are identified for the same items each
period.
• Is a product that has undergone a major quality
change the same product it was?
• A substantial quality improvement also may cause
an increase in the price of a product.
Slide 21
Quantity Indexes
An index that measures changes in quantity levels
over time is called a quantity index.
Probably the best known quantity index is the Index
of Industrial Production.
A weighted aggregate quantity index is computed in
much the same way as a weighted aggregate price
index.
A weighted aggregate quantity index for period t is
given by
Q it w i
It (100)
Qi0 wi
Slide 22
End of Chapter 20
Slide 23
