1.5: The National Debt…How Big Is It?
- Page ID
- 147902
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)INTRODUCTION
In this Collaboration, we will be discussing the concept of inflation. Inflation is the rate at which the general level of prices for goods and services is rising and, consequently, the buying power of currency is falling. In other words, if everything else (rent, groceries, utilities, etc.) gets more expensive, but your salary stays the same, your buying power decreases. To account for that, some state and private sector employees get annual salary adjustments based on the Consumer Price Index (CPI). Those adjustments are often referred to as Cost of Living Adjustment (COLA).
The following example illustrates the concept of adjusting for inflation. Consider a person who made $65,800 in 2022. How much would they need to make in 2023 to maintain the same buying power?
| Year | Annual Expenses | Annual Salary |
| 2022 | $20,000 | $65,800 |
| 2023 | $22,000 |
Another approach is to model this problem as a proportion and solve it for the missing quantity; Annual Salary in 2022 (denoted as x).
The reasoning behind this proportion is that in order to maintain the same buying power, the fraction of Annual Salary corresponding to Annual Expenses should be the same in 2022 and 2023.
\[ \dfrac{Annual\:Expenses\:in\:2022}{Annual\:Salary\:in\:2022} = \dfrac{Annual\:Expenses\:in\:2023}{Annual\:Salary\:in\:2023} \nonumber\]
\[ \dfrac{20,000}{65,800} = \dfrac{22,000}{x} \nonumber\]
\[ 20,000x = 22,000 \times 65,800 \nonumber\]
\[ x = 72,380 \nonumber\]
Check:
\[ \dfrac{20,000}{65,800} = \dfrac{22,000}{72,380} \nonumber\]
\[ 30.4\% = 30.4\% \nonumber\]
SPECIFIC OBJECTIVES
By the end of this collaboration, you should understand that
- proportions are one way to compare numbers of varying magnitudes.
- percentages are useful when making comparisons across time.
By the end of this collaboration, you should be able to
- use the Consumer Price Index and proportional reasoning to adjust dollar amounts for inflation.
- compare the U.S. national debt at various times in history by both 1) adjusting for inflation, and 2) expressing the debt as a percentage of the Gross Domestic Product.
PROBLEM SITUATION 1: HOW MUCH IS THAT CAR?
(1) The average price of a new car in 1945 was $1,020. The Consumer Price Index was 18.0 in 1945 and 271.0 in 2021. Use this information to answer the following question: $1,020 in 1945 has the same buying power as how many dollars in 2021?
PROBLEM SITUATION 2: IS THE U.S. NATIONAL DEBT OUT OF CONTROL?
(2) The table below lists the U.S. national debt at various times in history. We need to adjust these figures for inflation. Adjusting them will allow us to more accurately compare the sizes of these debts to one another.
(a) Use the Consumer Price Index21 to adjust the national debt for each year so that the debt is expressed in 2021 dollars. Enter the adjusted amounts in the table below. Round to two decimal places.
| Year | National Debt (in billions of dollars) | Consumer Price Index (average annual) as reported by BLS | National Debt Adjusted for Inflation (expressed in billions of 2021 dollars) |
| 1945 | 259 | 18.0 | |
| 1980 | 930 | 82.4 | |
| 1995 | 4,974 | 152.4 | |
| 2010 | 14,025 | 218.1 | |
| 2021 | 28,429 | 271.0 |
(b) When adjusted for inflation, in which year was the national debt largest?
(c) When adjusted for inflation, in which year was the national debt smallest?
(3) When communicating quantitative information, an infographic can be a useful tool. In order to help people understand the national debt adjusted to 2021 dollar amounts, we can draw bags of money to illustrate the change. If we use a 4 cm segment (4 bags of money) to represent the 1945 national debt adjusted for inflation (see figure below), how long should the segments be for each of the other years’ national debt after adjusting to 2021 amounts? Round your answers to the nearest tenth of a centimeter.
(a) How long should the segment be for 1980? Round to one decimal place.
(b) How long should the segment be for 1995? Round to one decimal place.
(c) How long should the segment be for 2010? Round to one decimal place.
(d) How long should the segment be for 2021? Round to one decimal place.
(4) According to the U.S. Treasury Department, a one-hundred-dollar bill is 0.0043 inches thick. Therefore, a stack of 100 one-hundred-dollar bills would be 0.43 inches thick. Assume that we could create stacks of one-hundred-dollar bills that would never topple over, no matter the height.
(a) How tall is the stack of money that represents the 1945 national debt, adjusted for inflation (expressed in billions of 2021 dollars)? Express your answer in miles and round to the nearest mile.
(b) How tall is the stack of money that represents the 2021 national debt? Express your answer in miles and round to the nearest mile.
PROBLEM SITUATION 3: ANOTHER VIEW OF THE U.S. NATIONAL DEBT?
(5) Now let us consider a different approach to analyzing the U.S. national debt over time. Just as the cost of dinner and a movie for Carlotta before and after college can be compared as a percentage of her income (as we thought about in the Preparation N.5), economists frequently compare the national debt at different times in history by measuring it as a percentage of Gross Domestic Product (GDP). The GDP is the estimated value of all the goods and services that are produced in a country in a given year.
(a) Fill in the table below with the national debt as a percentage of each year’s GDP. Round to the nearest percentage point.
| Year |
National Debt |
GDP (in billions of current dollars)22 | National Debt as a Percentage of GDP |
| 1945 | 259 | 228 | |
| 1980 | 930 | 2,857 | |
| 1995 | 4,974 | 7,640 | |
| 2010 | 14,025 | 15,049 | |
| 2021 | 28,429 | 23,315 |
(b) After adjusting the debt relative to GDP, in which year was the national debt the largest? In which year was the debt smallest? (Note: In this case, we say the national debts are being compared in relative terms.)
(c) Without adjustment, in which year was the national debt the largest? In which year was it the smallest? (Note: In this case, we say the national debts are being compared in absolute terms.)
(d) Is it better to measure the national debt in absolute terms or relative terms? Write a brief explanation.
MAKING CONNECTIONS
Record the important mathematical ideas from the discussion.
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