Wk8 DQ - Data Analysis & Business Intelligence

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Index Numbers

Chapter 17

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In this chapter, we will examine a useful descriptive tool called an index. An index expresses the relative change in a value from one period to another. Indexes are published on a regular basis by the federal government, by most business publications and daily newspapers, and on the Internet.

Learning Objectives

LO17-1 Compute and interpret a simple, unweighted index

LO17-2 Compute and interpret an unweighted aggregate index

LO17-3 Compute and interpret a weighted aggregate index

LO17-4 List and describe special-purpose indexes

LO17-5 Apply the Consumer Price Index

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Index Numbers

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 change in numbers

INDEX NUMBER A number that expresses the relative change in price, quantity, or value compared to a base period.

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An index number measures the relative change in a particular item (typically a product or service) between two time periods. If the index number is used to measure the relative change in just one variable, such as hourly wages in manufacturing, it is referred to as a simple index.

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?

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A simple price index is constructed by taking the price in a selected year and dividing it by the price in the base year and multiplying the result by 100. You can check the latest information on wages, the Consumer Price Index, and other business-related values at the Bureau of Labor Statistics (BLS) website, http://www.bls.gov

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?

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In this example, we are comparing the population of one province in Canada to the population of another Canadian province.

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 ÷ 42.9) × (100) = 224.2

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The chart reveals that in 2014, Miami and Charlotte served about the same number of passengers as Las Vegas (all three indexes are close to 100). Atlanta served more than twice as many passengers as McCarran International.

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

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Converting the data to an index and expressing as a percent increase makes it easier to access the trend; it is difficult to put the increase in perspective when just looking at the large numbers.

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

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In the formula, Pt is the price other than the base period and is often referred to as the given period or selected period. P0 is the base-period price.

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) ÷ 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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The base period need not be a single year. You can use more than one year as a base by averaging the prices of the selected years. In this example, a single year, 2005, two years 2005-2006, and three years 2005-2007 are used as the base periods.

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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There are two types of price indexes, unweighted and weighted. First, we’ll compute an unweighted price index. Earlier we calculated a simple index with formula 17-1 by comparing the base period price to a price in a given period. With formula 17-2 we can determine the percentage change in a group of items by averaging the simple indexes.

Simple Average of Price Indexes Example

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A positive feature of the simple average of price indexes is that we would obtain the same value for the index regardless of the unit of measure. A negative feature of this index is that it fails to consider the relative importance of the items included in the index. For example, milk and eggs receive the same weight, even though a typical family might spend far more over the year on milk than on eggs. Also, note that bacon is a significant contributor of the total index so the current price of bacon will drive the index more than any other item.

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

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Rather than summing the indexes, with this approach, we sum the prices for the two periods and then determine the index based on the price totals. Because the value of a simple aggregate index can be influenced by the units of measurement, it is not used frequently. We also need a way to weight the items according to their relative importance.

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

Advantage of the Laspeyres method

Only quantity data from the base period is used which allows for more meaningful comparison over time

Disadvantage

Does not reflect changes in buying patterns over time

It may overweight goods whose prices increase

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Now, weighted indexes are introduced; there are two methods of computing a weighted price index that differ only in the period used for weighting. The Laspeyres method uses base period quantities in both the base period and the given period. The base period weights, that is, the original prices and quantities of the purchased items, are used to find the percent change over a period of time in either price or quantity consumed, depending on the problem. Here is the table of the 6 food items used earlier (with quantities included) that will be used in the next example.

Laspeyres Price Index Example

We conclude the price of this group of items has increased 3.0% from 2009 to 2019.

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To determine the Laspeyres Price Index, we determine a weighted price index using the 6 food items from the previous example. This time we have prices and typical quantities consumed. The advantage of this method over the simple aggregate index is that the weight of each of the items is considered. In this index, milk is the item with the most weight (compared to bacon in the earlier example) because the product of the price and quantity is the largest.

Weighted Indexes: Paasche Method

In the Paasche method, current period quantities are used

Advantage of Paasche method

Current buying habits are reflected

Disadvantage

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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With the Paasche method, the current-period quantities are used for weights rather than the base-period quantities. The advantage of this is that we are using more recent quantities. P is the price index, pt is the current price, p0 is the price in the base period, qt is the quantity used in the current period.

Paasche Price Index Example

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Use the 6 food items used in earlier examples to determine the Paasche index. With this method, we find it costs 11.3% less to purchase these items in 2019 than it did in 2009.

Fisher’s Ideal Index

Fisher’s ideal index is the geometric mean of the Laspeyres and Paasche indexes

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Fisher’s ideal index combines the best features of the Laspeyres and Paasche methods and theoretically seems ideal; however, it is rarely used because it has the same problem that the Paasche index has. That is, you have to determine a new set of quantities for each period.

Value Index

A value index uses both base period and current period prices and quantities

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A value index measures changes in both the price and quantities involved. In the formula, P is the price index, Pt is the current price, p0 is the price in the base period, qt is the quantity in the current period, and q0 is the quantity in the base period. Using the data from table on the left, calculate the value index in the table on the right. The value index is 117.8; therefore, the value of apparel sales in 2019 was 117.8% of the 2015 sales, or 17.8% higher.

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 include department store sales (40%), regional employment (30%), freight car loadings (10%), and exports from Seattle harbor (20%).

Business activity has increased 57.0% from 2005 to 2010 and 57.1% from 2005 to 2018.

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The director assigns weights to each factor and using 2005 as the base year, expresses each 2010 and 2018 value as a percentage, with the base period value as the denominator. To illustrate, department store sales are converted to a percentage by ($44 ÷ $20)(100) = 220; this means department store sales increased 120% in the period. This percentage is then multiplied by the appropriate weight, in this example, (220)(.40) = 88.0. Do this calculation for the other factors and then sum the results. The 2010 index is 157.0 and 2018 is 157.1.

Consumer Price Index

The BLS reports this index monthly. It describes the changes in prices from one period to another for a “market basket” of goods and services.

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You can access this information by going to http://www.bls.gov, then under Data Tools select Inflation and Prices, then select All Urban Consumers (Current Series), Top Picks, and then click on U.S. All items 1982-84 = 100. You can select different time periods. The CPI will be discussed in more detail later in the chapter.

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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Formerly called the Wholesale Price Index, the Producer Price Index, PPI, dates back to 1890 and is published by the U.S. Bureau of Labor Statistics. To access this information, go to http://www.bls.gov, then Data Tools, select Inflation and Prices, then Commodity Data, Top Picks, and finally select Finished Goods. You can select different periods. The PPI measures inflation at earlier stages of the production and marketing process than the CPI.

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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You can find the current value of the DJIA by going to http://www.marketwatch.com. By clicking on the word “Dow” you can find additional details on the changes. At the bottom of the chart you can change the horizontal scale to show the changes for a day, 5 days, and so on up to 5 years. Here we are observing changes for a year. You can find other stock market indexes on the website too.

Consumer Price Index (CPI)

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 “real” income and purchasing power and to adjust pensions and tax brackets

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Originated in 1913, the CPI measures the change in price of a fixed market basket of goods and services from one period to another and is a widely reported index. The base period has to be updated occasionally because consumer’s purchasing patterns change. Since 1978, the BLS began publishing CPIs for two groups. One index, the Consumer Price Index – All Urban Consumers, covers about 87% of the total population. The other index, the Consumer Price Index – Urban Wage Earners and Clerical Workers, is a subset of the CPI-U and covers about 32% of the population. We’ll use the CPI to determine real income, deflate sales, and determine the purchasing power of the dollar.

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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The concept of real income is sometimes called deflated income or income expressed in constant dollars. In this example, to determine if Ms. Watts’s standard of living changed, her money income was converted to constant dollars. We find that her standard of living remained the same from the base period to the present time.

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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A price index can also be used to “deflate” sales or other similar money series. The PPI seems an appropriate index to use to deflate the manufacturer’s sales. Sales are listed in the second column, the PPI for each year is listed in the third column, the fourth column shows sales divided by the PPI. The right-hand column details the calculations. Sales (second column) increased each year. However, if we compare sales in constant dollars (fourth column) in 1982 and 2018, they are nearly the same.

Purchasing Power of the Dollar

Suppose the Consumer Price Index this month is 200.0 (1982-84 = 100). What is the purchasing power of the dollar?

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The Consumer Price Index is used to determine the purchasing power of the dollar. It is also used to make cost-of-living adjustments, or COLAs, in many management-union contracts; the specific clause in the contract is called the “escalator clause.” Social security benefits, child support, alimony, workers’ compensation, rental payments, and so on, are also adjusted with the CPI.

Chapter 17 Practice Problems

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Question 1

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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

LO17-1

Question 5

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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.

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.

LO17-1,2,3

Question 11

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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%.

Using 2000 as the base period, construct a leading economic indicator for 2018.

Interpret your leading index.

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Question 15

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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.

LO17-5