4.7: Corequisite- Buying Power, Consumer Price Index, Absolute and Relative Change, Reading and Interpreting Graphs
- Page ID
- 148610
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)SPECIFIC OBJECTIVES
By the end of this lesson, you should understand that
- ratios provide a way of comparing the relative increase or decrease of two variables.
- index numbers are a way of comparing the relative size of a variable over time.
By the end of this lesson, you should be able to
- use proportional reasoning to find the size of a variable that remains a constant proportion of another variable.
- use index numbers to find the value of a variable relative to time (or another variable).
PROBLEM SITUATION: THE BUYING POWER OF THE MINIMUM WAGE
The United States has a federal minimum wage. This means that there is a law that requires employers to pay employees a minimum amount. Many states also have their own minimum wage laws that require a higher wage than the federal law. The minimum wage started in 1938 as part of a law that protected the rights of workers in many ways. The minimum wage in 1938 was set at $0.25 per hour.19
Since 1938, Congress has increased the minimum wage many times to account for inflation. Inflation is when prices increase over time. Maybe you have heard stories from older people about how a cup of coffee used to cost a dime or a gallon of gasoline was less than a dollar. Prices for these items have increased due to inflation. Deflation is when prices decrease over time.
When deciding if the minimum wage should be increased, people often talk about the purchasing power of the wage. Did the minimum wage in 1938 have more purchasing power than the minimum wage in 2021? You will start by thinking about the specific example of buying a movie ticket for the following questions where you can compare the purchasing power of the minimum wage in 1997 to the minimum wage in 2021.
(1) The minimum wage in 1997 was $5.15 per hour. The minimum wage in 2021 was $7.25 per hour.
(a) What was the absolute change in the minimum wage from 1997 to 2021?
(b) What was the relative change in the minimum wage from 1997 to 2021? Round to the nearest tenth of one percent.
(2) The average price of a movie ticket in 1997 was $4.59.20 Which statement is correct? Be prepared to explain your answer.
(i) In 1997, a person earning minimum wage had to work less than an hour to earn enough for a movie ticket.
(ii) In 1997, a person earning minimum wage had to work more than an hour to earn enough for a movie ticket.
(3) The average price of a movie ticket in 2021 was $9.57.21 Which statement is correct? Be prepared to explain your answer.
(i) In 2021, a person earning minimum wage had to work less than an hour to earn enough for a movie ticket.
(ii) In 2021, a person earning minimum wage had to work more than an hour to earn enough for a movie ticket.
As we just explored, the minimum wage did not increase enough from 1997 to 2021 to keep up with the cost of a movie ticket. Remember that the minimum wage is the minimum amount employers must pay their employees required by federal law. A movie ticket is only a single product that someone might buy. To compare the purchasing power of the minimum wage from 1938 to 2021, you need more information about more products. You will use a tool called an index number.
Index numbers are a way to compare the relative change or difference in a data set, such as the prices of products. The change can be measured over time or over different geographic regions. There are different types of index numbers. One type is a measure of average relative change in data. In this instance, one data point is called the base and is assigned the value of 100 (meaning it represents 100%). The other figures are adjusted in proportion to the base. The plural form of the word index is indices.
The following two questions will help you understand index numbers. You will begin by looking at one product: the Big Mac.
(4) You are going to use 1992 as the base year for our Big Mac Price Index. Prices of the Big Mac vary with time. In 1992, a Big Mac cost about $2.19.
(a) In 1968, a Big Mac cost about $0.49. Compare the prices by finding the ratio of the 1968 price to the 1992 price. Express the ratio as a percent. Round to the nearest percent.
(b) In 2021, a Big Mac cost about $4.93.22 Compare the prices by finding the ratio of the 2021 price to the 1992 price. Express the ratio as a percent. Round to the nearest percent.
(c) Now let’s think about how to use your answers to (a) and (b) to complete the Big Mac Price Index for the years 1968, 1992, and 2021. In parts (a) and (b), what year did you use as the base? What is the index number for 1992 and 2021?
Big Mac Price Index
| Year | Index Number |
| 1968 | |
| 1992 | |
| 2021 |
The Value of the Dollar
Now you have seen how an index is created and what it means. You are now going to look at a real index that the U.S. government uses to track changes in the prices of many items over time. It is called the Consumer Price Index (CPI). You will use a version called the Urban Consumer Price Index (CPI-U).
The Bureau of Labor Statistics publishes the CPI-U each month. The CPI is a measure of the weighted average of a “basket of consumer goods and services.” The CPI-U basket includes transportation, food, medical care, housing, apparel, recreation, and education. The CPI represents the cost of that basket of goods and services in base-year dollars. This allows us to compare the buying power of a dollar in one year as compared to another. If the CPI is more than 100, then average prices are higher than in the base year. If the CPI is less than 100, then average prices are lower than in the base year. Because the CPI compares prices over time, it is one of the most frequently used statistics for identifying periods of inflation or deflation (increase or decrease of prices over time). An increase in the CPI indicates inflation in prices. A decrease indicates deflation. The following table shows the indices for selected years from 1913 to 2021.23
| Year | Index | Year | Index | Year | Index | ||
| 1913 | 9.9 | 1949 | 23.8 | 2005 | 195.3 | ||
| 1914 | 10.0 | 1950 | 24.1 | 2006 | 201.6 | ||
| 1915 | 10.1 | 1981 | 90.9 | 2007 | 207.3 | ||
| 1921 | 17.9 | 1982 | 96.5 | 2008 | 215.3 | ||
| 1922 | 16.8 | 1983 | 99.6 | 2009 | 214.5 | ||
| 1923 | 17.1 | 1984 | 103.9 | 2010 | 218.1 | ||
| 1924 | 17.1 | 1985 | 107.6 | 2012 | 229.6 | ||
| 1925 | 17.5 | 1996 | 156.9 | 2014 | 236.0 | ||
| 1936 | 13.9 | 1997 | 160.5 | 2015 | 237.0 | ||
| 1937 | 14.4 | 1998 | 163.0 | 2016 | 240.0 | ||
| 1938 | 14.1 | 1999 | 166.6 | 2017 | 245.1 | ||
| 1939 | 13.9 | 2000 | 172.2 | 2018 | 251.1 | ||
| 1940 | 14.0 | 2001 | 177.1 | 2019 | 255.7 | ||
| 1946 | 19.5 | 2002 | 179.9 | 2020 | 258.8 | ||
| 1947 | 22.3 | 2003 | 184.0 | 2021 | 271.0 | ||
| 1948 | 24.1 | 2004 | 188.9 |
(5) What year is used as the base for the CPI-U? What does the base mean?
(6) How does the 2021 dollar compare in value to the 1983 dollar?
(7) How does the 1949 dollar compare in value to the 1983 dollar?
(8) Who had more purchasing power? Provide quantitative information to support your answer.
- The person making minimum wage ($0.25 per hour) in 1938.
- The person making minimum wage ($7.25 per hour) in 2021.
Hint: What are some ways you could compare one index number (or wage) to another? How many working hours does it take in 1938 to buy one "basket of goods"? How many working hours does it take in 2021 to buy the same "basket of goods"?
(9) Find the price of gasoline in 1981 and in 2021. Give the sources for your information. Use the CPI-U table to evaluate the statement, “Gasoline was more expensive in 2021 than in 1981.” Provide mathematical information to justify your explanation.
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19 https://www.dol.gov/general/aboutdol/history/flsa1938