Statistics 222

1
Statistics 222

• Chapter 17
• Index Numbers

2
Indexes

• Each month, the U.S. Government publishes a
  variety of indexes designed to help individuals
  understand current business and economic
  conditions.
• The Consumer Price Index (CPI) is one such index.

3
The Consumer Price Index

• The CPI measures changes in price over a period
  of time.
• A year is selected to be the starting point or
  base period, and prices at that time are given an
  index of 100.
• Then all future years prices are expressed as a
  index relative to the base year.
• So if this years CPI is 125, then prices have
  risen approximately 25 since the base year.
• Lets discuss how this and some other indexes are
  calculated.

4
Indices that are Price Relatives

• The simplest price relative is based on the price
  of one item and its index expresses how the
  current price compares to the base period price.
• Lets take a gallon of gasoline. If the base year
  is 1984, and at that time a gallon of gas costs
  1.21, and if today a gallon of gas costs 2.79,
  then the price of a gallon of gas today can be
  expressed as the price relative of

2.79
(100) 230
1.21
5

• The price relatives for a gallon of gas using
  1984 as the base year is listed on the left.
• The price of a gallon of gas for any year can be
  compared easily to the base year. For example, in
  1997, the price was 1.7 higher than in 1984 but
  in 1998, the price was 12.4 lower.

6
Aggregate Price Indexes

• Aggregate price indexes express the relative
  price of a group (or basket) of items.
• For example, we could develop an aggregate price
  index for the cost of living, or the cost of
  owning a home, or owning a vehicle.

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Example of an aggregate price index the cost of
owning a car

• Lets assume there are four costs associated with
  owning a car the cost of gas, oil, tires, and
  auto insurance.
• We could calculate an unweighted price index or a
  weighted price index for that entire basket of
  items as a whole.

8
Unweighted price index for owning a car
1.36 2.20 145.00 700.00
I2002
1.21 1.50 80.00 300.00
222 (The cost of owning a car increased 122
from 1984 to 2002)
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The problem with using an unweighted price index

• As you can see the unweighted aggregate price
  index is calculated by simply summing the prices
  of the four items and then dividing that sum by
  the sum of the same four items prices in the
  base year.
• But that method gives more weight to the higher
  priced items even though their frequency of
  purchase may be much less than lower-priced
  items.
• A weighted aggregate price index assigns weights
  to the items based on their quantity of usage.

10
Weighted price index for owning a car
1.36(1000) 2.20(15) 145.00(2) 700.00(1)
I2002
1.21(1000) 1.50(15) 80.00(2) 300.00(1)
141 (The weighted cost of owning a car
increased 41 from 1984 to 2002)
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The weights

• The usage weights are based on mileage of 15,000
  miles per year tire usage assumes a 30,000 mile
  life.
• Clearly, the weighted index provides a more
  accurate indication of the change in costs of
  automobile ownership.

12
Laspeyres vs Paasche indexes

• When developing a weighted price index based on
  usage, the year-of-usage must be decided upon.
• Should we base usage on quantities in the base
  year or in the current year? (More fuel efficient
  cars would cause gas usage to change over time.)
• We could do it either way.
• If we use base-year usage figures, then we call
  it a Laspeyres index if we use current-year
  usage figures, then we call it a Paasche index.

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Example (p. 762 - 3)

• A large manufacturer purchases an identical
  component from three independent suppliers that
  differ in unit price and quantity supplied. The
  relevant data for 2001 and 2003 are given on the
  next slide.
• A. Compute the price relatives for each of the
  component suppliers separately. Compare the price
  increases by the suppliers over the two year
  period.
• B. Compute an unweighted aggregate price index
  for the component part in 2003.
• C. Compute a 2003 weighted aggregate price index
  for the component part. What is the
  interpretation of this index for the
  manufacturing firm?

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One item purchased from three different suppliers
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Question (a)

• Compute the price relatives for each of the
  component suppliers separately. Compare the price
  increases by the suppliers over the two year
  period.

6.00
Price relative for Supplier A
(100) 110.09

5.45
5.95
Price relative for Supplier B
(100) 106.25

5.60
6.20
Price relative for Supplier C
(100) 112.73

5.50
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Question (b)

• Compute an unweighted aggregate price index for
  the component part in 2003.

6.00 5.95 6.20
I2003
5.45 5.50 5.60
109.67
(The average cost of the item has increased 9.67
from 2001 to 2003)
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Question (c)

• Compute a 2003 weighted aggregate price index for
  the component part. What is the interpretation of
  this index for the manufacturing firm?

6.00 (150) 5.95 (200) 6.20 (120)
I2003
5.45 (150) 5.50 (200) 5.60 (120)
109.10
(The weighted average cost of the item has
increased 9.1 from 2001 to 2003)
18
The Consumer Price Index

• The Consumer Price Index published monthly by the
  U.S. Bureau of Labor Statistics, is the primary
  measure of the cost of living in the US.
• The group of items used to develop the index
  consists of a market basket of 400 items.
• The CPI is a weighted aggregate price index with
  fixed weights. The weights are derived from a
  usage survey.
• In Feb, 2003 the CPI (using 1982 to 1984 36
  month average) as the base year, was 183.1.

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The Consumer Price Index

• See Frequently Asked Questions
• http//www.bls.gov/cpi/cpifaq.htmQuestion207

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The Producer Price Index

• The PPI is also published monthly by the U.S.
  Bureau of Labor Statistics and measures the
  monthly changes in prices in primary markets in
  the U.S.
• The PPI is a family of indexes that measures the
  average change over time in selling prices
  received by domestic producers of goods and
  services. PPIs measure price change from the
  perspective of the seller. This contrasts with
  other measures, such as the Consumer Price Index
  (CPI), that measure price change from the
  purchaser's perspective.

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The Producer Price Index

• See Frequently Asked Questions
• http//www.bls.gov/ppi/ppifaq.htm1

22
The Dow Jones Industrial Average

• The DJIA is based on common stock prices of 30
  large companies. It is the sum of these stock
  prices divided by a number, which is revised from
  time-to-time to adjust for stock splits and
  switching of companies in the index.
• There are other Dow Jones industrial averages
  computed for specific industry sectors such as
  the transportation index and the utility index.

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The 30 companies in the DJIA
24
Dow Jones Industrial Average

• http//en.wikipedia.org/wiki/Dow_Jones_Industrial_
  Average

25
Deflating a Series by Price Index

• If you started a job with an annual salary of
  35,000 a year and two years later you were
  making 39,000 a year, what is your increase in
  real wages (purchasing power)?
• Your increase in salary was 11 but if the
  inflation rate was higher than 11, then you
  increase in real wages was a decrease, in other
  words, you lost purchasing power.

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When to deflate a series by a price index

• A economic report might show that the value of
  the inventory held by car dealerships is up 20.
  Does that mean that there are 20 more cars on
  the dealers lots?
• No, because some of that increase in inventory
  value is due to the rise in the price of cars.
• To find the real increase in inventory value, we
  must factor out the portion of that increase
  that is due to price increases.
• When we remove the the price increase effect, we
  are said to be deflating the increase.

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Using the CPI to deflate a series

• The CPI measures price increases in consumer
  goods.
• Purchasing power refers to how many/much consumer
  goods you can buy for your dollar.
• Real increases in wages refers to how many/much
  more goods you can buy because of your wage
  increase.
• Real increase in wages is the percentage increase
  with inflation factored out.
• To calculate your real wage increase, you must
  divide the overall percentage increase in wages
  by the CPI.

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This table shows the average hourly wage for all
U.S. workers for the years 1998 to 2002 and the
percentage change
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This table shows the average hourly wage for all
U.S. workers for the years 1998 to 2002 and the
percentage change after adjusting for inflation
(overall increases in the prices of consumer
goods)
Deflated hourly wage (hourly wage / CPI) 100