IndexNumbers.doc

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indicator not only of the current status of the stock market but of the economic well-being of the country as

a whole.

Although we can develop an index to measure changes in any time series, in practice most indices measure

changes in prices. Our notation will reflect this fact.

SECTION 24.2 Simple and Aggregate Index Numbers

We start with the most basic type of index number, the simple index.

Simple Index

A simple index is a ratio of the price of a commodity at time t divided by its value at some base period. We

can express the ratio as a percentage by multiplying by 100. That is,

100

0

1

1

=

P

PI

where

=1

I Index in the current period

=1

P Price in the current period

=0

P Price in the base period

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EXAMPLE 24.1

Construct the index of mean weekly earnings of all U.S. workers for 1990, 1995, and 1997, using 1985 as

the base year.

MEAN WEEKLY

YEAR EARNINGS

1985 $343

1990 412

1995 479

1997 503

SOURCE: U.S. Department of Labor, Bureau of Labor Statistics, Monthly Labor Reviews.

Solution

For each year, we compute

1000

1

1

= P

P

I

The results are as follows.

INDEX NUMBER

YEAR OF EARNINGS

1985 100

1990 120.1

1995 139.7

1997 146.6

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These numbers can be used, for example, by a labor union to compare its members' wage increases since

1985 with those of the average worker. Example 24.2 illustrates this type of comparison.

EXAMPLE 24.2

Suppose the members of a union were paid the mean weekly wages listed below. Compute the index of

earnings for 1990, 1995, and 1997 (using 1985 as the base year) for this union and compare it with the

index computed in Example 24.1.

MEAN WEEKLY EARNINGS

YEAR FOR A LABOR UNION

1985 $329

1990 387

1995 471

1997 508

Solution

The index of earnings for this union is as follows.

INDEX NUMBER

YEAR OF EARNINGS

1985 100

1990 117.6

1995 143.2

1997 154.4

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In comparison to the wages represented by the index numbers in Example 24.1, this union's wages grew

more slowly between 1985 and 1990 and more quickly between 1985 and 1995 and between 1985 and

1997.

The simple index number measures the price changes of only one item. Frequently though, we would like

to measure price changes for a group of commodities. The simple aggregate index number performs this

function.

Simple Aggregate Index

A simple aggregate index is the ratio of the sum of the prices of several commodities in the current period

to the sum in the base period, multiplied by 100.

100

1 0

1 1

1

=

=

=n

i i

n

i i

P

PI

where

iP1 = Price of item i in the current period

iP0 = Price of item i in the base period

EXAMPLE 24.3

Construct a simple aggregate index of the prices of meat, chicken, and fish items shown in the

accompanying table.

PRICE PER POUND

ITEM July 1998 July 1999

Beef $5.50 $5.92

Veal 7.48 8.06

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Pork 4.80 5.05

Chicken 3.62 3.45

Fish 5.13 6.25

Solution

100

1 0

1 1

1

=

=

=n

i i

n

i i

P

PI

= 10013.562.380.448.750.5

25.645.305.506.892.5

++++

++++= 108.3

Thus, the one-year increase in the total prices is 8.3%.

The figure calculated in Example 24.3 cannot be interpreted as the price increase for this part of our diets,

however. For example, if the average person's diet consisted mostly of chicken, the average costs might

actually have decreased. Thus, the simple aggregate index must be modified if we do not consume equal

quantities of each item. In the weighted aggregate index, each item is weighted by its relative importance.

Weighted Aggregate Index

100

01 0

1 11

1

=

=

=

i

n

i i

n

i ii

QP

QPI

where

iQ1 = Weight assigned to item i in the current period

iQ0 = Weight assigned to item i in the base period

Note that

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=n

i iQ

1 1= 1 and=

n

i iQ

1 0= 1

EXAMPLE 24.4

In July 1998, a family's weekly diet consisted of 6 pounds of fish, 2 pounds of beef, and 2 pounds of veal.

One year later, because of the cost increases in these products, the family's diet changed so that each week

they consumed 4 pounds of chicken and 1 pound of each other item. Assuming that the prices are those

listed in the table in Example 24.3, calculate the weighted aggregate index for 1999, using 1998 as the

base period.

Solution

The weight assigned to each food item in each year is the percentage of the entire diet supplied by that

item. The relevant weights are shown in the following table,

JULY 1998 JULY 1999

ITEM Price Quantity Weighting Price Quantity Weighting

Beef $5.50 2 0.2 $5.92 1 0.125

Veal 7.48 2 0.2 8.06 1 0.125

Pork 4.80 0 0.0 5.05 1 0.125

Chicken 3.62 0 0.0 3.45 4 0.500

Fish 5.13 6 0.6 6.25 1 0.125

Totals 10 1.0 8 1.0

The weighted aggregate index is

100

01 0

1 11

1

=

=

=

i

n

i i

n

i ii

QP

QPI

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1.86100)6(.13.5)0(62.3)0(80.4)2(.48.7)2(.50.5

)25.1(25.6)500(.45.3)125(.50.5)125(.06.8)125(.92.5 =

++++ ++++=

Thus, there has been a decrease of 13.9% in this part of the family's food budget. The evident problem with

this calculation, however, is that it does not reflect the price changes that have taken place.

One way of correcting the weighted aggregate index is to substitute the weights in the base period for the

weights in the current period. The result is the Laspeyres index.

Laspeyres Index

100

01 0

1 01

1

=

=

=

i

n

i i

n

i ii

QP

QPL

By keeping the weights the same, the Laspeyres index more accurately measures the true price changes, as

Example 24.5 illustrates.

EXAMPLE 24.5

Calculate the Laspeyres index for the family described in Example 24.4.

Solution

100

01 0

1 01

1

=

=

=

i

n

i i

n

i ii

QP

QPL

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4.115100)6(.13.5)0(62.3)0(80.4)2(.48.7)2(.50.5

)6(.25.6)0(45.3)0(50.5)2(.06.8)2(.92.5=

++++++++

=

This number tells us that, if the 1999 diet had been the same as the 1998 diet, this part of the family budget

would show a cost increase of 15.4%. Of course, since in 1999 the family no longer eats 6 pounds of fish, 2

pounds of beef, and 2 pounds of veal weekly, this figure does not accurately reflect the situation,

Another way of measuring the price increase is to use the current year's weighting. When this is done, the

result is the Paasche index,

Paasche Index

100

11 0

1 11

1

=

=

=

i

n

i i

n

i ii

QP

QPP

EXAMPLE 24.6

Calculate the Paasche index for the family in Exercise 24.4.

Solution

100

11 0

1 11

1

=

=

=

i

n

i i

n

i ii

QP

QPp

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5.104100)125(.13.5)500(.62.3)125(.80.4)125(.48.7)125(.50.5

)125(.25.6)500(.45.3)125(.50.5)125(.06.8)125(.92.5=

++++++++

=

Using the 1999 quantities, the aggregate price increase is 4.5%.

We now have three different indices based on the same data. The weighted aggregate index (from Example

24.4) is1

I = 86.1. The Laspeyres index (from Example 24.5) is1

L = 115.4. The Paasche index (from

Example 24.6) is1

P = 104.5.

Which index should we use to describe the price change? To answer this question, we must first address

another question: what do we want to measure? Do we want to find out what actually happened to this

family's food budget between 1998 and 1999? If so, we use1

I = 86.1; the cost decreased by 13.9%. Do

we want to know what happened to the prices, assuming that the 1998 diet remained unchanged? Then

we use1

L = 115.4; prices increased by 15.4%. Do we want to know what happened to the prices,

assuming that the 1999 diet was in effect in 1998? Then we use1

P = 104.5; prices increased by only

4.5%. Thus, our choice of index depends on what we want to measure.

Governments in the United States, Canada, and other countries have chosen to use the Laspeyres index to

measure how prices change monthly. The Consumer Price Index is an example of a Laspeyres index, and it

is without doubt the most important measure of inflation in many countries. It is important for managers

and economists to understand this descriptive measurement.

Consumer Price Index

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The Consumer Price Index works with a basket of some 300 goods and services in the United States (and a

similar number in other countries), including such diverse items as food, housing, clothing, transportation,

health, and recreation. Prices for each item in this basket are computed on a monthly basis, by city, state,

and region, and the CPI is computed from these prices.

The basket is defined for the "typical" or "average" middle-income family, and the set of items and their

weights are revised periodically (every 10 years in the United States and every 7 years in Canada).

There are, of course, a number of problems associated with the construction and the continuing validity of

an index such as the CPI. In constructing the index, the following steps must be performed:

1. Select the appropriate items for the basket.

2. Select the appropriate weights.

3. Select an appropriate base period.

For example, starting the index during a year of low prices or in the midst of a recession will cause a later

economic recovery to appear to be imposing substantial increases on the price index.

To maintain the validity of the CPI over a number of years, the index's sponsors must deal with several

problems. Has the typical family changed from the currently used moderate-income urban couple? How

much distortion occurs during the years between official revisions of the CPI (at which time new

consumption patterns are incorporated into the new index)? Have definitional changes occurred with

respect to terms such as "food away from home"? How important are qualitative changes, and does the

revised CPI capture these differences? For example, computers are more powerful today than last year and

last year's computers were more powerful than those produced the previous year. And yet, the price has

decreased.

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The CPI, despite never really being intended to serve as the official measure of inflation, has come to be

interpreted in this way by the general public. Pension-plan payments, old-age security, and some labor

contracts are automatically linked to the CPI and automatically indexed (so it is claimed) to the level of

inflation. However, if yours is not the "typical family," price changes may affect you quite differently from

the way suggested by the CPI.

Using the Consumer Price Index to Deflate Prices

Despite its flaws, the Consumer Price Index is used in numerous applications. One application involves

adjusting prices by removing the effect of inflation.

To do this, we first identify the CPI from 1980 to 1997 (see Table 24.1). The base year of the index is the

average of 1982-84. We can use this table to deflate the annual values of prices or wages. This removes the

effect of inflation, making comparisons more realistic. For example, suppose a worker earned $6.50/hour in

1985 and $9.25/hour in 1997. To determine whether his or her purchasing power really increased, we

deflate both figures by dividing each by the CPI for the corresponding year and then multiplying by 100.

Thus, we have the following figures:

DEFLATED

YEAR WAGE CPI WAGE

1985 $6.50 107.6 $6.04

1997 9.25 160.5 5.76

The deflated wages are now being measured in 1982-84 dollars. In 1982-84 dollars, the worker earned less

in 1997 than he or she did in 1985.

Another way of making such comparisons is by dividing the 1997 wage by the 1997 CPI and then

multiplying by the 1985 CPI:

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20.66.1075.160

25.9 =

This figure represents the worker's wages measured in 1985 dollars. Because in 1985 he or she actually

earned $6.50, we can see that our earlier conclusion remains unchanged.

TABLE 24.1 United States Consumer Price Index, 1980-1997

Year CPI Year CPI

1980 82.4 1989 124.0

1981 90.9 1990 130.7

1982 96.5 1991 136.2

1983 99.6 1992 140.3

1984 103.9 1993 144.5

1985 107.6 1994 148.2

1986 109.6 1995 152.4

1987 113.6 1996 156.9

1988 118.3 1997 160.5

SOURCE: U.S. Department of Commerce, Bureau of the Census, Statistical Abstract of the United States.

1998.

EXAMPLE 24.7

The gross domestic product (GDP) is often used as a measure of the economic growth of a country. The

annual GDP of the United States for the years 1990-1997 is shown in the accompanying table.

YEAR GDP ($billions)

1990 5743.8

1991 5916.7

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1992 6244.4

1993 6558.1

1994 6947.0

1995 7265.4

1996 7636.0

1997 8079.9

SOURCE: U.S. Department of Commerce, Bureau of the Census, Statistical Abstract of the United States:

1998.

Use the CPI in Table 24.1 to deflate these figures to 1982-84 dollars.

Solution

To convert the GDP to 1982-84 (sometimes referred to as "constant 1982-84") dollars, we divide the GDP

by its associated CPI and then multiply by the CPI in1982-84 (which is 100). The results follow.

YEAR GDP ($billions) GDP ($billions)

YEAR CURENT DOLLARS 1982-84 CONSTANT DOLLARS

1990 5743.8 4394.6

1991 5916.7 4344.1

1992 6244.4 4450.7

1993 6558.1 4538.5

1994 6947.0 4687.6

1995 7265.4 4767.3

1996 7636.0 4866.8

1997 8079.9 5034.2

As you can see, the GNP in current dollars gives the impression that the economy has grown rapidly in the

period 1990-1997, whereas when measured in 1982-84 constant dollars the growth is quite modest. Our

conclusion would be the same if we used some year other than 1982-84 as our basis.

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SECTION 24.3 Summary

We have discussed the basic concept of an index, and through successive examples we have shown what

several different indices measure. In particular, we noted that index numbers measure the changes over

time of particular time-series data. Simple indices measure changes in only a single data series, while

aggregate indices measure changes in several variables. We examined simple aggregate indices and

weighted aggregate indices. Two specific examples of the latter are the Laspeyres index and the Paasche

index. The most commonly used Laspeyres index is the Consumer Price Index. We discussed how the CPI

is determined, and we showed how it could be used to deflate prices in order to facilitate comparisons.

Important Terms

Index numbers

Simple index

Simple aggregate index

Weighted aggregate index

Laspeyres index

Paasche index

Consumer Price Index

Deflate

EXERCISES

24. 1 Taking 1990 as the base year, compute the simple index for the price of natural gas in 1989-1996.

Natural Gas Price

Year (per 1,000 cubic feet)

1989 $1.53

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1990 1.55

1991 1.48

1992 1.57

1993 1.84

1994 1.67

1995 1.40

1996 2.03

SOURCE: U.S, Department of Commerce, Bureau of the Census, Statistical Abstract of the United States.

1998.

24.2 Taking 1989 as the base year, calculate the simple index for the price of a short ton of coal in

1989-1996.

Coal Price

Year (per short ton)

1989 $1.84

1990 1.75

1991 1.61

1992 1.52

1993 1.46

1994 1.60

1995 1.76

1996 1.80


1998.

24.3 Repeat Exercise 24. 1, taking 1993 as the base year.

24.4 A cake recipe calls for the ingredients below. Compute a simple aggregate index, taking 1990 as the

base year.

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Price

Ingredient 1990 1998

Butter (per pound $1.27 $1.87

Sugar (per pound) .65 .45

Flour (per pound) 1.47 1.89

Eggs (per dozen) .52 .85

24.5 The hotel industry is very interested in understanding how tourists spend money. In order to measure

the price changes in three important components of a tourist's budget, a statistician computed the average

cost of a hotel room (one night), a meal, and a car rental lone day) in 1990 and in 1998. The results of these

computations are shown in the accompanying table.

Cost

Component 1990 1998

Hotel (one night) $75 $180

Meal 12 16

Car rental (one day) 18 40

Compute a simple aggregate index, taking 1990 as the base year.

24.6 Refer to Exercise 24.4. Suppose the chef at Chez Gerard's has decided to make his own improvements

(adding more butter) to the cake recipe he used in 1990. The old and new quantities of ingredients are listed

in the accompanying table. Compute a weighted aggregate index to measure the price increases.

1990 1998

Ingredient Price Quantity Price Quantity

Butter $1.27/lb 16 oz $1.87/lb 20 oz

Sugar .65/Ib 8 oz .95/lb 80 oz

Flour 1.49/lb 18 oz 1.89/lb 18 oz

Eggs .52/dozen 3eggs .85/dozen 3eggs

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24.7 Refer to Exercise 24.5. Suppose that in 1990 the average tourist stayed in the hotel for 6 days, ate 8

meals at the hotel, and rented a car for 2 days. In 1998, the average tourist stayed for 4 days, ate 6 meals at

the hotel, and rented a car for 3 days. Calculate a weighted aggregate index.

24.8 For Exercise 24.7, compute the Laspeyres index

24.9 For Exercise 24.7, compute the Paasche index.

24.10 For Exercise 24.6, calculate the Laspeyres index to measure the change in the cost of a cake's

ingredients.

24.11 For Exercise 24.6, calculate the Paasche index to measure the change in the cost of a cake's

ingredients.

24.12 What is being measured by each of the indices computed in Exercises 24.6, 24.10, and 24.11?

24.13 People continually look for investments in periods of high and unexpected inflation, buying gold,

silver, and even platinum, as a hedge against inflation becomes attractive to some investors. Actual prices

for these three precious metals are listed below. Compute the simple index for each metal for the years

1992, 1994, and 1997, taking 1989 as the base year.

Gold Price Silver Price Platinum Price

Year (per fine ounce) (per fine ounce) (per troy ounce)

1989 $383 $5.50 $507

1990 385 4.82 467

1991 363 4.04 371

1992 345 3.94 360

1993 361 4.30 374

1994 385 5.29 411

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1995 386 5.15 425

1996 389 5.19 398

1997 333 4.90 397


1998.

24.14 For Exercise 24.13 calculate the aggregate index for all three metals for 1997, taking 1989 as the

base year.

24.15 Suppose an investor in 1989 owned 100 ounces of gold, 1,000 ounces of silver, and 50 ounces of

platinum. By 1997, she had 500 ounces of gold, 1,500 ounces of silver, and 100 ounces of platinum.

Construct a weighted aggregate index to measure how her investment's value changed between 1989 and

1997.

24.16 For Exercise 24.15, compute the Laspeyres index.

24.17 For Exercise 24.15, compute the Paasche index.

24.18 The annual price of a barrel of oil for the years 1989-1997 is shown below. Calculate a simple index

showing the price increases for 1990, 1993, and 1997. Use 1989 as the base year.

Crude Petroleum Price

Year (per barrel)

1989 $15.86

1990 20.03

1991 16.54

1992 15.99

1993 14.25

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1994 13.19

1995 14.62

1996 18.46

1997 17.24


1998.

24. 19 Using Table 24. 1, deflate the annual prices of crude petroleum in Exercise 24.18 so that they are

measured in constant 1982-84 dollars.

Exercises 24.20 through 24.24 are based on the following problem.

A gasoline service station determined the price and the number of units sold per day of its four most

popular items (after gasoline). These data were recorded for the years 1985, 1990, and 1995, as shown

below. Taking 1985 as the base year, calculate the simple aggregate index for 1990 and 1995.

1985 1990 1995

Item Price Quantity Price Quantity Price Quantity

Oil (quart) $0.65 23 1.20 10 $1.85 5

Tire 23.00 12 55.00 15 75.00 16

Antifreeze (quart) .80 7 2.00 20 2.50 14

Battery 27.00 13 45.00 22 65.00 20

24.21 Compute the weighted aggregate index for 1990 and 1995, taking 1985 as the base year.

24.22 Compute the Laspeyres index for 1990 and 1995.

24.23 Compute the Paasche index for 1990 and 1995.

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24.24 Deflate (to 1982-84 constant dollars) each of the indices in Exercises 24.20 through 24.23, using the

CPI in Table 20.1. What conclusions can you reach about these results?

24.25 Is real estate a good investment? In particular, will the price of a house keep up with inflation? To

answer this question, the median sales price of new privately owned, one-family houses was recorded for

1989-1997. These data are shown next. Compute the prices in 1982-84 constant dollars, using the CPI in

Table 24.1. What conclusions do these results lead to?

Median Sales Price of New

Year One-family Houses ($000)

1989 120

1990 122.9

1991 120

1992 121.5

1993 126.5

1994 130

1995 133.9

1996 140

1997 146

SOURCE; U.S. Department of Commerce, Bureau of the Census, Statistical Abstract of the United States

1998.

24.26 The median sales price of existing one-family homes is listed next. Repeat Exercise 20.25 using these

data.

Median Sales Price of Existing

Year One-family Houses ($000)

1989 93.1

1990 95.5

1991 100.3

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1992 103.7

1993 106.8

1994 109.9

1995 113.1

1996 118.2

1997 124.1

SOURCE; U.S. Department of Commerce, Bureau of the Census, Statistical Abstract of the United States

1998.

24.27 The table that follows lists the per capita gross national product. Convert these figures to constant

1982-84 dollars. What do these values tell you?

Year Per Capita GNP

1989 21,984

1990 22,979

1991 23,416

1992 24,447

1993 25,403

1994 26,647

1995 27,605

1996 28,752

1997 30,161

SOURCE: U.S. Bureau of Economic Analysis, National Income and Products Accounts of the United

States, July 1998.

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