Statistical techniques in business ecohomics chap018

An Index Number expresses the relative change in price, quantity, or value compared to a base period... Types of Index NumbersIndexes: Four classifications Price Quantity Value Special p

Eighteen

McGraw-Hill/

Irwin

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

Chapter Eighteen

Index Numbers

GOALS

When you have completed this chapter, you

will be able to:

ONE

Describe the term index.

TWO

Understand the difference between a weighted price index and

an unweighted price index.

THREE

Construct and interpret a Laspeyres Price index.

FOUR

Chapter Eighteen continued

Index Numbers

GOALS

When you have completed this chapter, you

will be able to:

FIVE

Construct and interpret a Value Index.

SIX

Explain how the Consumer Price index is constructed and

interpreted.

Goals

Index Numbers

A Simple Index Number

measures the relative change in

just one variable

An Index Number

expresses the relative

change in price,

quantity, or value

compared to a base

period

36-Month CPI 2000-2002

0 1 2 3 4

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

Month beginning 1/1/2002

Stock 1997

Price

1997 Shares

2002 Price

2002 Shares

NWS $1 30 $2 50

NPC $5 15 $4 30

GAC $6 40 $6 20

Mr Wagner

owns stock in

three companies

Given is the

price per share

at the end of

1997 and 2002

for the three

stocks and the

quantities he

owned in 1997

and 2002.

Simple indexes using 1997 as

base year (1997=100)

Price

($2/$1)(100)=200 ($4/$5)(100)=80 ($6/$6)(100)=100

Share

(50/30)(100)=167 (30/15)(100)=200 (20/40)(100)=50

Example 1

Why Convert Data to Indexes?

CPI

Easier to

comprehend

than actual

numbers

(percent

change)

Why compute indexes?

Facilitate

comparison of

unlike series

Provide convenient ways to express the change in the total

of a heterogeneous group of items

Indexes

Bread $0.89

Car $18,000

Dress $200

Surgery $400,000

$345,651,289,560

or 10% ?

Types of Index Numbers

Indexes: Four classifications

Price

Quantity

Value

Special purpose

Measures the changes

in prices from a

selected base period

to another period

Measures the changes in quantity consumed from the base period to

another period

Measures the change in the

value of one or more items

from the base period to the

given period (PxQ).

Combines and weights a heterogeneous group of series

to arrive at an overall index showing the change in

business activity from the base period to the present

Producer Price Index - measures the

average change in prices received in the primary markets of the US by producers of commodities in all stages of processing (1982=100)

Price Index

Quantity Chicago Midwest Manufacturing Index Base year 1997=100

0 20 40 60 80 100 120 140 160 180

YEAR 1996 1997 1998 1999 2000 2001 2002

CFMMI CFMMI Auto

Federal Reserve

Quantity Output

Price and Quantity Indexes

Value Index of Feb '03 Retail Sales Base February (Monthly Average of

Oct 1999-Sept 2000)=100

0 10 20 30 40 50 60 70 80 90 100

Value

Special purpose

Value and Special Purpose Indexes

Construction of Index Numbers

P p

p

t



0

100 ( )

where

p o the base period price

p t the price at the selected

or given period

aggregate price index for

the three stocks

0 100

) 100

( 6

$ 5

$ 1

$

6

$ 5

$ 2

$

) 100

(

0



















p p

Two methods of

computing the

price index

the quantities of items

Laspeyres method

Paasche method where

period

Laspeyres Weighted Price Index, P

Uses the base period quantities as

weights

Tends to overweight goods whose prices have increased

p q

t



0

0 0

100

Construction of Index Numbers

p q

t

Paasche Weighted Price Index, P

Present year weights

substituted for the original

base period weights

Tends to overweight goods whose prices have

gone down

where

qt is the current quantity consumed

p0 is the price in the base period

pt is the current price

Fisher’s ideal index = (Laspeyres’ index)(Paasche’s index)

Balances the negative effects of the

Laspeyres’ and Paasche’s indices

Requires that a new set of quantities be determined each year.

The geometric

mean of Laspeyres

and Paasche

indexes

Fisher’s Ideal Index

Fisher’s Ideal Index

Value Index

V p q

p q

t t

 

 0 0 ( 100 )

Both the price and quantity change from the base period to the given period Reflects changes in both price and quantity

Value Index

Consumer Price Index

In 1978 two consumer price indexes were published One was designed for urban wage

earners and clerical workers

Millions of employees

in automobile, steel, and other industries have their wages adjusted upward when the CPI increases

It covers about one third of the

population Another was

designed for all urban

households It covers about

80% of the population

Consumer Price Index

Usefulness of CPI

It allows consumers to

determine the effect of

price increases on their

purchasing power.

It is a

yardstick for

revising

wages,

pensions,

alimony

payments, etc.

It is an economic indicator of the rate of inflation in the United States.

It computes real income:

real income

= money income/CPI

(100)

Consumer Price Index

Deflating Sales

Deflated sales Actual sales

An approximate index

Determining the purchasing power of the dollar

compared with its value for the base period

Purcha g power of dollar

CPI

sin  $1 ( 100 )

18- 18

Consumer Price Index

Shifting the base

When two or more series

of index numbers are to be compared,they may not have the same base period 101 115 First select a common base period for all series

101 115

Then use the respective base numbers as the denominators

and convert each series to the new base period

Mr Wagner

owns stock in

three companies

Shown below is

the price per

share at the end

of 1997 and

2002 for the

three stocks and

the quantities he

owned in 1997

and 2002

Stock 1997

Price

1997 Shares

2002 Price

2002 Shares

NWS $1 30 $2 50

NPC $5 15 $4 30

GAC $6 40 $6 20

35 104 )

100

( 345

$

360

$

) 100

( ) 40 ( 6

$ ) 15 ( 5

$ ) 30 ( 1

$

) 40 ( 6

$ ) 15 ( 4

$ ) 30 ( 2

$

) 100

(

0 0 0





















q p

q

p

Laspeyres Weighted Price Index, P

Example 1 continued

25 106 )

100

( 320

$

340

$

) 100

( ) 20 ( 6

$ ) 30 ( 5

$ ) 50 ( 1

$

) 20 ( 6

$ ) 30 ( 4

$ ) 50 ( 2

$

) 100

(

0





















t

t

t q p

q

p

P

F = (104.35)(106.25)

=105.3

Fisher’s Ideal Index

55 98 )

100

( 345

$

340

$

) 100

( ) 40 ( 6

$ )

15 ( 5

$ )

30 ( 1

$

) 20 ( 6

$ ) 30 ( 4

$ ) 50 ( 2

$

) 100

(

0 0





















q p

q

p

Value Index