{"id":604,"date":"2026-06-07T14:07:26","date_gmt":"2026-06-07T14:07:26","guid":{"rendered":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/?post_type=back-matter&p=604"},"modified":"2026-06-08T14:44:22","modified_gmt":"2026-06-08T14:44:22","slug":"alt-text-long-description","status":"publish","type":"back-matter","link":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/back-matter\/alt-text-long-description\/","title":{"raw":"Alt Text Long Description","rendered":"Alt Text Long Description"},"content":{"raw":"Figure 2.6<\/a>:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Distance Run<\/th>\r\nDistance Swim<\/th>\r\nTime to Victim<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0<\/td>\r\n141.4213562<\/td>\r\n70.71067812<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n137.9311422<\/td>\r\n69.96557112<\/td>\r\n<\/tr>\r\n
10<\/td>\r\n134.5362405<\/td>\r\n69.26812024<\/td>\r\n<\/tr>\r\n
15<\/td>\r\n131.2440475<\/td>\r\n68.62202374<\/td>\r\n<\/tr>\r\n
20<\/td>\r\n128.0624847<\/td>\r\n68.03124237<\/td>\r\n<\/tr>\r\n
25<\/td>\r\n125<\/td>\r\n67.5<\/td>\r\n<\/tr>\r\n
30<\/td>\r\n122.0655562<\/td>\r\n67.03277808<\/td>\r\n<\/tr>\r\n
35<\/td>\r\n119.2686044<\/td>\r\n66.63430221<\/td>\r\n<\/tr>\r\n
40<\/td>\r\n116.6190379<\/td>\r\n66.30951895<\/td>\r\n<\/tr>\r\n
45<\/td>\r\n114.1271221<\/td>\r\n66.06356105<\/td>\r\n<\/tr>\r\n
50<\/td>\r\n111.8033989<\/td>\r\n65.90169944<\/td>\r\n<\/tr>\r\n
55<\/td>\r\n109.658561<\/td>\r\n65.8292805<\/td>\r\n<\/tr>\r\n
60<\/td>\r\n107.7032961<\/td>\r\n65.85164807<\/td>\r\n<\/tr>\r\n
65<\/td>\r\n105.9481005<\/td>\r\n65.97405025<\/td>\r\n<\/tr>\r\n
70<\/td>\r\n104.4030651<\/td>\r\n66.20153254<\/td>\r\n<\/tr>\r\n
75<\/td>\r\n103.0776406<\/td>\r\n66.53882032<\/td>\r\n<\/tr>\r\n
80<\/td>\r\n101.9802903<\/td>\r\n66.99019514<\/td>\r\n<\/tr>\r\n
85<\/td>\r\n101.1187421<\/td>\r\n67.55937104<\/td>\r\n<\/tr>\r\n
90<\/td>\r\n100,4987562<\/td>\r\n68.24937811<\/td>\r\n<\/tr>\r\n
95<\/td>\r\n100.124922<\/td>\r\n69.06246099<\/td>\r\n<\/tr>\r\n
100<\/td>\r\n100<\/td>\r\n70<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Figure 2.9<\/a>: Scatter plot graph titled with axis labels \"Time to Victim (seconds)\" on the vertical axis (ranging from 55 to 75 seconds) and \"Distance Run (meters)\" on the horizontal axis (ranging from 0 to 120 meters). The data is plotted as blue dots forming a smooth U-shaped curve, with two orange highlighted data points labeled \"Initial\" and \"New.\" The curve begins at approximately 71 seconds at 0 meters run, decreases steadily to a minimum of approximately 59\u201360 seconds around 55\u201360 meters, then rises again back toward 70 seconds at 100 meters. This U-shape illustrates that there is an optimal distance to run along the shore before entering the water to minimize total rescue time, consistent with an optimization problem involving two different travel speeds (running vs. swimming). The \"Initial\" orange data point is marked at approximately 55 meters run and 66 seconds, indicating a starting guess or pre-optimization estimate. The \"New\" orange data point is marked at approximately 80 meters run and 59 seconds, indicating the improved or solver-optimized solution that minimizes time to reach the victim.<\/p>\r\n

Figure 2.10<\/a>: The diagram shows a horizontal baseline with several labeled reference points and a large curved arc above it representing the objective function. The horizontal axis contains four key labeled positions from left to right: \"Initial value\" (the starting point of the solver), a series of intermediate black dots representing successive iterations of the solver's path, \"Exact answer\" (the true mathematical optimum, marked with a dashed vertical line), and \"Solver's answer\" (the solver's final computed result, marked with a dashed vertical line slightly to the right of the exact answer). A large smooth arc labeled \"Objective function\" spans the full width of the diagram, peaking above the exact answer location, representing the function being maximized or minimized. A smaller nested arc labeled \"Solver's path\" illustrates the trajectory of the solver's iterative search process, showing how the solver takes progressively smaller steps, depicted as a series of diminishing arcs and dots along the baseline, as it homes in on the optimum. The solver's final answer lands slightly past the exact answer, illustrating that numerical solvers approximate but may not land precisely on the true optimum.<\/p>\r\n

Figure 2.11<\/a>: Screenshot of a Microsoft Excel spreadsheet with an embedded scatter plot chart and an open Solver Parameters dialog box, illustrating a profit maximization problem. Column A is labeled \"x\" and contains values from 0 to 5 in increments of 0.5. Column B is labeled \"profits\" and contains corresponding dollar values: $0.00 (x=0), $1.75 (x=0.5), $3.00 (x=1), $3.75 (x=1.5), $4.00 (x=2), $3.75 (x=2.5), $3.00 (x=3), $1.75 (x=3.5), $0.00 (x=4), -$2.25 (x=4.5), -$5.00 (x=5). The profit values form an inverted U-shape, peaking at x=2. Rows 17 and 18 show the solver's current variable value of approximately 1.999999995 for x and 4 for profits, representing the solver's optimized result. Scatter plot chart: Titled \"profits,\" the chart plots profits (vertical axis, ranging from -$6.00 to $5.00) against x units sold (horizontal axis, ranging from 0 to 6). The data points form a clear downward-opening parabolic curve, confirming a maximum profit around x=2. Solver Parameters dialog box: The objective cell is set to $A$18 (profits), the optimization direction is set to Max (selected), and the variable cell to be changed is $A$17 (the x value). The dialog is partially visible, cut off below the \"By Changing Variable Cells\" field.<\/p>\r\n

Figure 3.3<\/a>: Screenshot of a Monte Carlo simulation output page from an Excel add-in, consisting of two components: a summary statistics table and a histogram. Summary Statistics table (top) is a bordered table with two columns labeled \"Summary Statistics\" (blue header) and \"Notes\" (red header). The Notes column is empty. The statistics reported are: Average = 89.955, SD (standard deviation) = 3.1523, Max = 98.000, Min = 78.000. Histogram (bottom) is titled \"Histogram of $C$1.\" The chart displays the frequency distribution of simulation results as a bar chart with blue outlined bars. The horizontal axis runs from 78 to 98. The vertical axis is unlabeled. The distribution is approximately bell-shaped and roughly symmetrical, centered around the mean of approximately 90. The bars rise gradually from the left tail starting near 78, reach their peak height around values of 91\u201392, then taper off toward the right tail ending near 98.<\/p>\r\n

Figure 3.10<\/a>: Line chart titled \"Infection Rate = 5.0%\" (displayed in red at the top). The vertical axis is labeled \"Number of Tests\" ranging from 0 to 1000, and the horizontal axis is labeled \"Group Size\" ranging from 0 to 20. Four lines are plotted. Individual Tests (red horizontal line) show a flat line at 1000 tests, representing the baseline cost of testing every individual separately, regardless of group size. E[#Total Tests] (black curve) shows the expected total number of tests required under a group testing strategy. This curve forms a U-shape, starting high at small group sizes (approximately 600 at group size 2), decreasing to a minimum of approximately 420 at group size 5, then rising again as group size increases, reaching approximately 680 at group size 20. This U-shape identifies the optimal group size that minimizes total tests. E[#Pos Group Tests] (orange curve) shows the expected number of follow-up individual tests required for positive groups. This curve starts low (approximately 100 at group size 2) and increases steadily and smoothly as group size grows, reaching approximately 640 at group size 20, reflecting that larger groups are more likely to contain at least one infected individual. #Groups Tested (blue curve) show the number of initial group tests administered. This curve starts high (approximately 500 at group size 2) and decreases steadily as group size increases, reaching approximately 50 at group size 20, since fewer but larger groups are needed to cover the population. A vertical annotation on the horizontal axis marks GS=5* as the optimal group size that minimizes the total expected number of tests at the given 5.0% infection rate.<\/p>\r\n

Figure 3.11<\/a>: Line chart titled \"Infection Rate = 1.0%\" (displayed in red at the top) illustrating pooled\/group testing optimization at a lower infection rate than a companion 5.0% chart. The vertical axis is labeled \"Number of Tests\" ranging from 0 to 1000, and the horizontal axis is labeled \"Group Size\" ranging from 0 to 20. Four lines are plotted: Individual Tests (red horizontal line) shows a flat line at 1000 tests, representing the baseline cost of testing every individual separately. E[#Total Tests] (black curve) shows the expected total number of tests under a group testing strategy. The curve starts high at approximately 520 at group size 2, decreases steeply to a shallow minimum of approximately 180\u2013190 around group size 11, then rises very gradually toward approximately 240 at group size 20. The curve is much flatter near its minimum compared to the 5.0% infection rate chart, indicating less sensitivity to group size choice near the optimum. E[#Pos Group Tests] (orange curve) shows the expected number of follow-up individual tests for positive groups. This curve starts near 0 at small group sizes and increases slowly and steadily, reaching approximately 190 at group size 20, reflecting that at a 1.0% infection rate, groups are much less likely to test positive. #Groups Tested (blue curve) shows the number of initial group screens administered. This curve starts high at approximately 500 at group size 2 and decreases smoothly, reaching approximately 50 at group size 20. A dashed vertical line and annotation mark GS=11* as the optimal group size minimizing total expected tests at the 1.0% infection rate.<\/p>\r\n

Figure 3.12<\/a>: Monte Carlo simulation output page consisting of a summary statistics table and a histogram. Summary Statistics table (top) is a bordered table with two columns labeled \"Summary Statistics\" (blue header) and \"Notes\" (red header). The Notes column contains the annotation \"n = 5\" in red, indicating the sample size used in the simulation. The statistics reported are: Average = 0.367, SD = 0.1411, Max = 1.052, Min = 0.200. Histogram (bottom) is titled \"Histogram of $D$4.\" The chart displays the frequency distribution of simulation results as a step-style bar chart with blue outlined bars. The horizontal axis ranges from approximately 0.2 to 1.0+, representing the range of simulated output values. The vertical axis shows frequency counts (unlabeled). The distribution is strongly right-skewed (positively skewed), with the tallest bars concentrated at the left end near the minimum value of 0.2, and a long, gradually tapering tail extending to the right toward values above 1.0. The mode is near 0.2 and frequency drops off steeply and then more gradually as values increase.<\/p>\r\n

Figure 3.13<\/a>: Monte Carlo simulation output page from an Excel add-in, comparing two simulation results side by side using a summary statistics table and an overlaid histogram. Summary Statistics table (top) has three sections. The left section (pink header) shows statistics for cell $D$4: Average = 0.367, SD = 0.1414, Max = 1.110, Min = 0.200. The center section (blue header) shows statistics for DGP!$F$4: Average = 0.361, SD = 0.1638, Max = 1.107, Min = 0.160. The right \"Notes\" column (red text) states: \"n= 5 (pink) v 4 (blue) with 100,000 reps\" and \"n=4 is optimal.\" Histogram (bottom) is titled \"Histogram of $D$4 And DGP!F$4.\" The chart overlays two step-style frequency distribution curves. The pink line represents $D$4 (n=5) and the blue line represents DGP!F$4 (n=4). The horizontal axis ranges from approximately 0.16 to above 0.96. Both distributions are strongly right-skewed, with peaks near the left end and long tails extending to the right. The pink curve (n=5) peaks slightly higher and is positioned slightly to the right of the blue curve (n=4), which starts at a lower minimum value of 0.16 compared to 0.20 for n=5. The two distributions are otherwise closely overlapping throughout most of their range. A legend in the lower right identifies the two series. The comparison illustrates the difference in the distribution of the optimization metric between the two candidate group sizes.<\/p>\r\n

Figure 3.14<\/a>: Combined spreadsheet table and line chart illustrating an optimal search theory model showing expected total cost as a function of the number of searches for a single search cost scenario. Spreadsheet table (left): header text states \"Prices U[0,1], q=1 and c=0.04,\" indicating prices are uniformly distributed between 0 and 1, quantity equals 1, and search cost c=0.04. The table has four columns: Number of Searches (1\u201310), Expected Value of Price, Search Cost, and Expected Value of Total Cost. Values begin at 1 search with expected price = 0.500, search cost = 0.04, and total cost = 0.540, then decline as searches increase. Row 4 is highlighted in blue bold italic, showing: 4 searches, expected price = 0.200, search cost = 0.16, total cost = 0.360, identifying n*=4 as the optimal number of searches. Values beyond row 4 show total cost rising again: row 5 total cost = 0.367, continuing upward to 0.491 at 10 searches. The line chart (right) is titled \"Expected Value of Total Cost =f(search).\" The vertical axis is labeled \"expected value of total cost\" ranging from 0.000 to 0.600, and the horizontal axis is labeled \"n (number of searches)\" ranging from 0 to 12. A single blue U-shaped curve is plotted, starting high at approximately 0.540 at n=1, declining steeply to a minimum at approximately 0.360, then rising gradually back toward 0.500 at n=10. A dashed vertical blue line marks the minimum at n=4*, identifying the optimal number of searches that minimizes total expected cost at this search cost level of c=0.04.<\/p>\r\n

Figure 3.15<\/a>: Combined spreadsheet table and line chart illustrating an optimal search theory model comparing expected total cost as a function of the number of searches under two different search cost scenarios. Spreadsheet table (left): Header text states \"Prices U[0,1], q=1 and c=0.04,\" indicating prices are uniformly distributed between 0 and 1, quantity equals 1, and search cost c=0.04. The table has four columns: Number of Searches (1\u201315), Expected Value of Price, Search Cost, and Expected Value of Total Cost. Key values include: at 1 search, expected price = 0.500, search cost = 0.01, total cost = 0.510; costs decline as searches increase to a minimum; row 9 is highlighted in red bold italic, showing: 9 searches, expected price = 0.100, search cost = 0.09, total cost = 0.190, identifying n*=9 as the optimal number of searches for c=0.04. Values beyond row 9 show total cost beginning to rise again. Line chart (right): Titled \"Expected Value of Total Cost =f(search) for c=0.04 and c=0.01.\" The vertical axis is labeled \"expected value of total cost\" ranging from 0.000 to 0.600, and the horizontal axis is labeled \"n (number of searches)\" ranging from 0 to 16. Two curves are plotted: a blue curve labeled \"TC c=0.01\" (lower search cost) and a red curve labeled \"TC c=0.04\" (higher search cost). Both curves are U-shaped, declining steeply at first then rising gradually. The blue curve sits higher overall due to lower per-search cost allowing more searches, with its minimum marked by a dashed blue vertical line at n=4*. The red curve sits lower overall and reaches its minimum at a dashed red vertical line at n=9*. The chart illustrates that lower search costs lead to fewer optimal searches while higher search costs lead to more, consistent with optimal search theory.<\/p>\r\n

Figure 4.2<\/a>: Line chart titled \"Racing Sequences\" comparing the growth trajectories of arithmetic and geometric sequences over time. The vertical axis shows dollar values ranging from $0 to $600,000,000, and the horizontal axis shows days ranging from 0 to 230. The Arithmetic sequence (blue line) is a straight, steadily increasing diagonal line that grows linearly from $0 at day 0 to approximately $200,000,000 by around day 200, illustrating constant additive growth at a fixed amount per day. The geometric sequence (orange dotted line) is a curve that remains nearly flat and close to zero for the majority of the time period, virtually indistinguishable from the baseline until approximately day 175\u2013180, at which point it begins to curve sharply upward in a classic exponential growth pattern. By approximately day 200 it overtakes the arithmetic sequence, and by approximately day 215 it reaches nearly $500,000,000, far exceeding the arithmetic sequence's value. The dotted styling of the geometric line may indicate projected or extrapolated values beyond a certain point.<\/p>\r\n

Figure 4.4<\/a>: Line chart comparing average income or wealth in the United States and Japan from approximately 1955 to 2022, measured in thousands of euros at 2022 purchasing power parity (PPP). The vertical axis shows values in thousand euros (2022) ranging from 0 to 70k, and the horizontal axis shows years from approximately 1955 to 2022. Two lines are plotted. The US line (red) begins at approximately \u20ac24,000\u201325,000 in the mid-1950s and rises steadily with periodic fluctuations throughout the entire period. Growth accelerates notably from the 1980s onward, reaching approximately \u20ac55,000 by 2000 and peaking near \u20ac68,000\u201369,000 around 2021\u20132022, with a brief dip around 2008\u20132009 consistent with the global financial crisis. The Japan line (blue) begins at approximately \u20ac5,000 in the mid-1950s and grows rapidly through the 1970s and 1980s, reflecting Japan's postwar economic miracle. Growth slows considerably after approximately 1990, consistent with Japan's \"Lost Decade,\" with the line flattening and plateauing around \u20ac33,000\u201338,000 through the 1990s and 2000s, before modest growth resumes toward approximately \u20ac37,000 by 2022.<\/span><\/p>\r\n

Figure 4.6<\/a>: Line chart showing college wage premium as a percentage for different racial and ethnic groups in the United States from 2000 to approximately 2022. The vertical axis is labeled \"Wage premium (%)\" ranging from 60 to 130, and the horizontal axis shows years from 2000 to 2022. Five lines are plotted, each representing a different demographic group. The Asian line (green) is the highest line throughout the entire period, beginning at approximately 82\u201383% in 2000 and rising with considerable year-to-year volatility to a peak of approximately 125% around 2019\u20132020, before declining to approximately 112% by 2022. The overall upward trend is strong and the Asian wage premium is substantially higher than all other groups throughout the period. Black (blue line) begins at approximately 76% in 2000, fluctuates moderately, reaches a peak of approximately 90% around 2013, then gradually declines toward approximately 72\u201375% by 2022, ending close to the overall average. Overall (red line, thick) is the bold reference line representing the average wage premium across all groups. Begins at approximately 69% in 2000 and rises gradually to approximately 77\u201378% by the early 2020s, with modest year-to-year variation. White (yellow\/orange line) tracks closely below the overall average throughout the period, beginning at approximately 67\u201368% in 2000 and rising modestly to approximately 72\u201374% by 2022. Hispanic (dark red\/brown line) is the lowest line for most of the period, beginning at approximately 68% in 2000, fluctuating between approximately 62% and 79%, and ending at approximately 70% by 2022, slightly below the overall average.<\/span><\/p>\r\n

Figure 4.7<\/a>: Line chart titled \"Hypothetical Projected HS versus College Income\" comparing cumulative or annual income trajectories for high school (HS) and college graduates over a working lifetime. The vertical axis shows dollar values ranging from -$50,000 to $200,000, and the horizontal axis shows age ranging from 18 to 68. Two curves are plotted: College line (orange curve) begins below zero at age 18, dipping into negative territory (toward -$50,000) to represent the cost of college tuition and foregone income during the college years. The curve rises steeply from approximately age 22 onward, crossing into positive income territory and continuing to grow at a diminishing rate, reaching approximately $185,000 by age 68. HS line (blue curve) begins at approximately $35,000 at age 18, representing immediate entry into the workforce after high school. The curve rises gradually and at a diminishing rate throughout the career, reaching approximately $110,000 by age 68. Three labeled regions illustrate the key financial components of the college investment decision: Negative A (marked near age 18 on the horizontal axis) represents the direct cost of college tuition and expenses, shown as the negative dip in the orange curve below zero. Negative B (marked with a dashed vertical orange line near age 22\u201323) represents the opportunity cost of college, the foregone income a college student gives up compared to a high school graduate working during those years. +C (labeled in the large region between the two curves during the working years) represents the cumulative earnings premium that college graduates earn over high school graduates throughout their career, which offsets the earlier costs A and B. The chart is a standard educational illustration of the human capital investment model, demonstrating that college has upfront costs (-A and -B) but generates long-term income benefits (+C) that, if sufficiently large, justify the investment.<\/span><\/p>\r\n

Figure 4.8<\/a>: Scatter plot chart titled \"HS and College Income Streams\" comparing the annual income of high school (HS) and college graduates as discrete yearly data points across a working lifetime. The vertical axis shows dollar values ranging from -$30,000 to $80,000, and the horizontal axis shows age ranging from 18 to 68. Two data series are plotted as dotted lines of individual points. College (orange dots) show ages 18, 19, 20, and 21. The college data points appear at approximately -$20,000, representing net negative income during the college years due to tuition costs and foregone earnings. A dashed vertical orange line marks approximately age 22, indicating the transition from college to workforce entry. From age 22 onward, the orange dots form a flat horizontal line at exactly $70,000 per year, continuing consistently through age 65\u201368, representing a constant annual college graduate salary.\r\nHS (blue dots) begin at age 18. The blue dots form a flat horizontal line at exactly $40,000 per year, continuing consistently through age 65\u201368, representing a constant annual high school graduate salary throughout the entire working career with no interruption.<\/span><\/p>\r\n

Figure 5.3<\/a>: Line chart titled \"Unemployment Rate\" showing the U.S. unemployment rate as a percentage from January 1952 to approximately 2022. The vertical axis shows percent ranging from 0.0 to 16.0, and the horizontal axis shows dates at approximately decade intervals from Jan-52 to Jul-13 and beyond. Vertical gray shaded bands mark periods of U.S. economic recessions throughout the chart. The unemployment rate begins at approximately 3% in early 1952 and fluctuates cyclically throughout the entire period, with recurring peaks during recessions followed by recoveries to lower levels. Notable peaks include: approximately 6% in the mid-1950s recession; approximately 7\u20137.5% during the early 1960s recession; approximately 9% during the mid-1970s recession following the oil shock; a dramatic peak of approximately 10.8% in late 1982\u20131983 during the severe double-dip recession; approximately 7.8% in the early 1990s recession; approximately 6.3% in the early 2000s following the dot-com bust; approximately 10% in 2009\u20132010 during the Great Recession; and a catastrophic spike to approximately 14.7% in April 2020 due to the COVID-19 pandemic shutdown, the highest value in the entire series, followed by an extremely rapid recovery back to approximately 3.5\u20134% by 2022.<\/span><\/p>\r\n

Figure 5.4<\/a>: Line chart titled \"Labor Force Participation Rate\" showing the percentage of the U.S. civilian noninstitutional population participating in the labor force from January 1952 to approximately 2022. The vertical axis shows percent ranging from 52.0 to 68.0, and the horizontal axis shows dates at approximately decade intervals. Vertical gray shaded bands mark periods of U.S. economic recessions. The chart reveals four distinct phases. 1952\u2013early 1960s (flat\/slightly declining) shows the participation rate hovers in a narrow band between approximately 58.5% and 60.0%, with modest cyclical fluctuations around recessions but no clear trend. Mid-1960s through 2000 (long sustained rise) shows that the rate rises steadily and substantially from approximately 58.5% in the mid-1960s to a peak of approximately 67.3% around 2000, an increase of nearly 9 percentage points over roughly 35 years. The ascent is nearly continuous, interrupted only briefly by recessions, and accelerates noticeably through the 1970s and 1980s. 2000\u20132015 (prolonged decline) shows that following the peak around 2000, the participation rate declines persistently from approximately 67.3% to approximately 62.4% by 2015. The decline accelerates during and after the Great Recession (2008\u20132009). 2015\u20132022 (stabilization then COVID shock) shows the rate stabilizes around 62.5\u201363.5% before plunging sharply to approximately 60.2% in April 2020 due to the COVID-19 pandemic, then recovering partially to approximately 62.5% by 2022, remaining well below pre-pandemic and historical peak levels.<\/span><\/p>\r\n

Figure 5.6<\/a>: Line chart titled \"Male and Female Unemployment Rate\" comparing unemployment rates for women and men aged 20 and over in the United States from January 1948 to approximately 2022. The vertical axis shows percent ranging from 0.0 to 18.0, and the horizontal axis shows dates at approximately decade intervals from Jan-48 to Jan-13 and beyond. Two lines are plotted: Female 20+ (orange line) represents the unemployment rate for adult women aged 20 and older. Male 20+ (green line) represents the unemployment rate for adult men aged 20 and older. From 1948\u2013late 1960s, the two lines track closely together with moderate cyclical fluctuations, both ranging between approximately 2.5% and 7\u20138%. Female unemployment is occasionally slightly higher than male during this period. From 1970s\u20131980s, both series rise substantially, reflecting the stagflation era and recessions. Female unemployment peaks at approximately 8.5% and male at approximately 6% during the mid-1970s recession. During the severe 1981\u20131982 recession, male unemployment surges dramatically to approximately 10% while female peaks slightly lower at approximately 9.5%, marking one of the most notable periods where male unemployment exceeds female. From 1990s\u20132000s, the two lines converge and track very closely together, both cycling between approximately 3.5% and 7\u20138% through successive recessions, with female unemployment generally slightly above or equal to male. From 2008\u20132010 (Great Recession), male unemployment rises more sharply than female, peaking at approximately 10.5% compared to approximately 8% for females. 2020 (COVID-19 pandemic) shows both series spike dramatically. Female unemployment surges to approximately 15.5% while male reaches approximately 11.5%, with female unemployment exceeding male by the largest margin in the entire series. Both recover rapidly to approximately 3.5% by 2022.<\/span><\/p>\r\n

Figure 5.7<\/a>: Line chart titled \"Overall and Teenage Unemployment Rate\" comparing the overall U.S. unemployment rate with the teenage unemployment rate (ages 16\u201319) from January 1948 to approximately 2022. The vertical axis shows percent ranging from 0.0 to 35.0, and the horizontal axis shows dates at approximately decade intervals from Jan-48 onward. Two lines are plotted: Overall (blue line) represents the total U.S. unemployment rate across all age groups, tracking closely with the series shown in companion charts. The line fluctuates cyclically between approximately 2.5% and 10.8%, with recession-era peaks visible throughout the period and a dramatic spike to approximately 14.7% in April 2020 during the COVID-19 pandemic, before rapidly recovering to approximately 3.5% by 2022. Teenage 16\u201319yrs (green line) represents the unemployment rate for teenagers aged 16\u201319. This line runs consistently and substantially above the overall rate throughout the entire 74-year period, typically at a level two to three times higher than the overall rate at any given time. Key observations include: the teenage rate begins at approximately 9\u201315% in the late 1940s and early 1950s; it rises with each recession, peaking at approximately 24% during the severe 1981\u20131982 recession; it reaches approximately 27% during the Great Recession of 2009\u20132010; and it spikes to approximately 32.5% during the COVID-19 pandemic in 2020 before recovering sharply to approximately 10\u201311% by 2022. <\/span><\/p>\r\n

Figure 5.8<\/a>: Line chart titled \"Unemployment Rate by Race and Ethnicity\" comparing U.S. unemployment rates across five racial and ethnic groups from 1948 to approximately 2022. The vertical axis shows percent ranging from 0.0 to 25.0, and the horizontal axis shows years from 1948 to 2028. A legend identifies five series. Black (gray line) shows consistently the highest unemployment rate throughout the entire period. Begins at approximately 7\u20138% in the late 1940s and fluctuates cyclically, reaching a dramatic peak of approximately 21% during the 1981\u20131982 recession \u2014 the highest value of any group in the series. Subsequent recession peaks occur at approximately 16\u201317% in the early 1990s and approximately 16% during the Great Recession of 2009\u20132010. The COVID-19 pandemic spike reaches approximately 15\u201316% before recovering to approximately 5\u20136% by 2022. Throughout the entire period, Black unemployment remains roughly double the White rate. Hispanic (blue line) shows data beginning in the mid-1970s. It consistently tracks above the overall and White rates but below the Black rate for most of the period. It peaks at approximately 15.5% during the 1981\u20131982 recession and approximately 13% during the Great Recession. The COVID-19 spike reaches approximately 18.5%, briefly exceeding the Black rate, before recovering to approximately 4\u20135% by 2022. Overall (black line) is the reference series representing all workers combined, tracking between the higher-unemployment minority groups and the lower White rate. Cyclical peaks reach approximately 10\u201311% during the 1981\u20131982 recession and approximately 10% during the Great Recession, with a COVID spike to approximately 14.7%. White (orange line) is consistently among the lowest unemployment rates throughout the period, tracking closely below the overall rate. It peaks at approximately 9\u201310% during the 1981\u20131982 recession and approximately 9% during the Great Recession, with a COVID spike to approximately 14% before rapid recovery to approximately 3.5%. Asian (green line) has data beginning in the early 2000s and is available only for the most recent portion of the chart. It tracks at or below the White and overall rates during normal periods, reaching approximately 3\u20134% at cyclical lows. It experiences a sharp COVID-19 spike to approximately 15% in 2020 before recovering rapidly to approximately 3% by 2022.<\/span><\/p>\r\n

Figure 5.10<\/a>: Line chart comparing average Not Seasonally Adjusted (NSA) and Seasonally Adjusted (SA) unemployment or economic data by month across a full calendar year. The vertical axis shows values ranging from 0 to 7, and the horizontal axis shows all twelve months from January through December. A \"Month\" dropdown filter button is visible at the bottom left, and toggle buttons for \"Average of NSA\" and \"Average of SA\" appear at the top left. Two lines are plotted: Average of NSA (blue line) represents the raw, not seasonally adjusted average values by month. The line shows clear seasonal variation, beginning high at approximately 6.3 in January, remaining elevated at approximately 6.4 in February, then declining through spring to a trough of approximately 5.5 in April\u2013May, rising modestly to approximately 6.0 in June, then declining again through summer and fall to a low of approximately 5.2 in November, before recovering slightly to approximately 5.5 in December. The blue line oscillates noticeably throughout the year, reflecting true seasonal patterns in the underlying data. Average of SA (orange line) represents the seasonally adjusted average values by month. The line is nearly flat throughout the entire year, hovering consistently between approximately 5.6 and 5.8 with minimal month-to-month variation. This flat profile illustrates the effect of seasonal adjustment, which removes predictable seasonal fluctuations to reveal the underlying trend.<\/span><\/p>\r\n

Figure 6.3<\/a>: Three-dimensional surface chart titled \"Utility Maximization\" displaying a utility function over two consumption goods. The three axes are: Utility (vertical axis, ranging from approximately -120 to 130), Brandies (horizontal axis along the left base, ranging from 0 to approximately 10), and Cigars (horizontal axis along the right base, ranging from approximately 3 to 18). The surface forms a smooth, dome-shaped hill or inverted paraboloid, rising from low or negative utility values at the edges and corners of the consumption space to a clear single peak or maximum in the interior of the surface. The peak appears to occur at intermediate values of both brandies and cigars, at a utility level of approximately 120\u2013130. Beyond the optimal combination in any direction utility declines, eventually falling into negative values at the extremes of the axes. The surface is rendered with a multi-colored gradient from blue and green at the lower utility regions through yellow, orange, and red toward the peak, providing a visual heat-map-like indication of utility levels across the consumption space.<\/span><\/p>\r\n

Figure 6.5<\/a>: Contour plot titled \"Utility Maximization: Contour Plot\" displaying a top-down view of the utility surface from the companion three-dimensional surface chart. The horizontal axis is labeled \"Brandies\" ranging from 0 to 10, and the vertical axis is labeled \"Cigars\" ranging from 0 to 20. The plot uses a multi-colored concentric ring pattern to represent utility levels across all combinations of the two goods, with each colored band representing a contour line of equal utility (an indifference curve). The color gradient transitions from warm colors (orange and red) at the outermost rings representing lower utility levels, through yellow and green in the intermediate rings, to cool blue and light blue at the innermost rings representing the highest utility levels, converging toward a single optimal point at the center of the innermost ring. A red dot marks the utility-maximizing consumption bundle, and an annotation box connected to it by a line reads: \"B=3, C=10 yields U*=127,\" identifying the optimal solution as 3 brandies and 10 cigars, which produces a maximum utility value of 127. The contour plot provides a two-dimensional representation of the same utility surface shown in the 3D chart, making it easier to identify the precise location of the utility maximum and visualize how utility changes as consumption of either good moves away from the optimal bundle in any direction.<\/span><\/p>\r\n

Figure 6.6<\/a>: Side-by-side pair of charts titled \"Constrained Utility Maximization: 3D Surface\" (left) and \"Constrained Utility Maximization: Contour Plot\" (right), extending the unconstrained utility maximization charts by adding a budget constraint. The left panel is three-dimensional surface chart identical in structure to the unconstrained utility surface, with axes for Utility (vertical, approximately -120 to 130), Brandies (horizontal base axis, 0 to approximately 10), and Cigars (horizontal base axis, approximately 3 to 18). The same dome-shaped utility surface is displayed with a multi-colored gradient. A red planar surface cuts through the utility dome diagonally, representing the budget constraint \u2014 a flat plane that intersects the utility surface and limits the feasible consumption combinations to those at or below the budget. The constrained optimum occurs at the point where the budget constraint plane is tangent to the highest reachable point on the utility surface. The right panel is a top-down contour plot with the same concentric multi-colored rings as the unconstrained version, with Brandies on the horizontal axis (0 to 10) and Cigars on the vertical axis (0 to 20). A red diagonal line represents the budget constraint, running from Cigars, 5 to Brandies, 5 of the feasible consumption space, indicating the trade-off between purchasing brandies and cigars given a fixed budget. A light blue dot marks the constrained utility-maximizing consumption bundle on the budget line, and an annotation box reads: \"B=1, C=4 yields U*=79,\" identifying the optimal constrained solution as 1 brandy and 4 cigars, yielding a maximum attainable utility of 79, substantially lower than the unconstrained maximum of 127, illustrating the utility cost of the budget constraint.<\/span><\/p>\r\n

Figure 6.7<\/a>: Wireframe contour plot titled \"Constrained Utility Maximization: Wireframe Contour\" displaying the same utility maximization problem as the companion filled contour plot, but rendered as a wireframe (outline-only) version without color fill. The horizontal axis is labeled \"Brandies\" ranging from 0 to 10, and the vertical axis is labeled \"Cigars\" ranging from 0 to 20. The wireframe consists of concentric oval\/elliptical contour lines drawn in varying colors (blue, orange, yellow, green, and red) representing indifference curves of equal utility at different levels. The contours are centered around the unconstrained utility maximum located at approximately Brandies=3, Cigars=10, with inner rings representing higher utility levels and outer rings representing lower utility levels. The elliptical shape of the contours indicates that the utility function values the two goods differently, with the contours being more elongated along the Cigars axis than the Brandies axis. A red diagonal straight line runs from approximately the point (0, 5) on the Cigars axis down to approximately (5.5, 0) on the Brandies axis, representing the budget constraint. The line indicates the maximum affordable combinations of brandies and cigars given the consumer's budget and the prices of each good. The constrained optimum occurs where the budget constraint line is tangent to the highest reachable indifference curve, which based on the companion contour plot is at B*=1, C*=4 yielding U*=79.<\/span><\/p>\r\n

Figure 6.8<\/a>: Small diagram illustrating the tangency condition for constrained utility maximization, showing three indifference curves and a budget constraint line. Three blue curved lines arc from the upper left to the lower right, representing indifference curves at three distinct utility levels, labeled on the right side as: U>79 (uppermost curve, highest utility level), U=79* (middle curve, the optimal utility level), and U<79 (lowest curve, lowest utility level shown). A red diagonal straight line represents the budget constraint, running from upper left to lower right across all three curves. The red budget constraint line intersects the lowest indifference curve (U<79) at two points, crosses through the middle curve (U*=79) at exactly one point of tangency, and does not reach the highest indifference curve (U>79), which lies entirely beyond the budget constraint.<\/span><\/p>\r\n

Figure 6.10<\/a>: Line chart titled \"B* = f(Total)\" showing the relationship between total budget allowed and the optimal consumption of brandies in a constrained utility maximization model. The horizontal axis is labeled \"Total Allowed\" ranging from 0 to 12, representing the consumer's total budget or spending limit. The vertical axis is labeled \"Optimal Brandies Consumption\" ranging from 0 to 2.5, representing the utility-maximizing quantity of brandies given each budget level. A single blue line with data point markers connects seven plotted points, beginning at approximately (5, 1.0) and rising steadily and nearly linearly to approximately (10, 2.25). The data points and approximate values are: Total=5 \u2192 B*\u22481.00; Total=6 \u2192 B*\u22481.25; Total=7 \u2192 B*\u22481.50; Total=8 \u2192 B*\u22481.75; Total=9 \u2192 B*\u22482.00; Total=10 \u2192 B*\u22482.25. No data points are plotted below a total budget of 5.<\/span><\/p>\r\n

Figure 6.11<\/a>: Horizontal number line diagram illustrating the concept of elasticity in economics, ranging from negative infinity (\u2212\u221e) on the left to positive infinity (+\u221e) on the right, with key values marked at \u22122, \u22121, 0, +1, and +2. The diagram is divided into three conceptual zones with labels above the number line. The left side (negative values) are labeled \"Negative Inverse relationship,\" with two downward-pointing arrows indicating that when the elasticity is negative, the two variables move in opposite directions (one rises while the other falls). A leftward arrow labeled \"Responsiveness Increasing\" runs from 0 toward \u2212\u221e, indicating that as the elasticity value becomes more negative, the responsiveness or sensitivity of the relationship increases. The center (zero) is labeled \"No relationship,\" with a downward arrow pointing to 0 on the number line, indicating that an elasticity of zero means the dependent variable does not respond at all to changes in the independent variable. The right side (positive values) is labeled \"Positive Direct relationship,\" with two upward-pointing arrows indicating that when the elasticity is positive, the two variables move in the same direction. A rightward arrow labeled \"Responsiveness Increasing\" runs from 0 toward +\u221e, indicating that as the elasticity value becomes more positive, responsiveness increases. Below the number line, five key elasticity benchmarks are labeled at their corresponding positions: Perfectly Elastic (at \u2212\u221e), Unit Elastic (at \u22121), Perfectly Inelastic (at 0), Unit Elastic (at +1), and Perfectly Elastic (at +\u221e).<\/span><\/p>\r\n

Figure 6.12<\/a>: Line chart titled \"Sales of cigarettes per adult per day.\" The subtitle notes that figures include manufactured cigarettes as well as estimated hand-rolled cigarettes per adult aged 15 and over per day. The vertical axis shows cigarette quantities from 0 to approximately 11 cigarettes, and the horizontal axis shows years from approximately 1900 to 2015. Two lines are plotted. United States (red line) begins near 0 cigarettes per adult per day around 1900 and rises steeply and steadily through the early twentieth century, reaching approximately 2 cigarettes by 1920. Growth continues through the 1930s and accelerates sharply in the 1940s and 1950s, peaking at approximately 10.5\u201311 cigarettes per adult per day around 1963\u20131965, the highest value in the entire chart. Following the peak, the US line declines persistently and steadily through the 1970s, 1980s, 1990s, and 2000s, reaching approximately 3 cigarettes per day by 2015. Japan (blue line) data begins in the early 1900s at low levels, tracking similarly to the US through the 1920s and 1930s at approximately 2 cigarettes. A sharp dip occurs around 1945\u20131946, dropping to approximately 1 cigarette. Japan's consumption rises steeply through the 1950s, 1960s, and 1970s, peaking at approximately 9.5\u201310 cigarettes per adult per day around 1977\u20131980, roughly a decade after the US peak. Japan's subsequent decline is slower and more gradual than the US, reaching approximately 4.5 cigarettes per day by 2015, remaining substantially higher than the US at the end of the series.<\/span><\/p>\r\n

Figure 6.14<\/a>: Line chart illustrating a cost minimization problem in microeconomics or operations management, showing an isoquant curve and three isocost lines. The vertical axis ranges from 0 to 100,000 and the horizontal axis ranges from 0 to 100,000, with both axes representing quantities of two inputs (unlabeled). Isoquant curve (thick dark blue curve) is a downward-sloping convex curve labeled \"q=300.\" The curve begins at approximately (1,000, 90,000) in the upper left and sweeps downward to near (100,000, 0) at the lower right, exhibiting the typical diminishing marginal rate of technical substitution shape. Three isocost lines (thin straight diagonal blue lines): Three parallel downward-sloping straight lines representing different total cost levels, labeled on the right side of the chart: TC = $2M (uppermost line, highest cost), TC = $1.2M (middle line), and TC = $0.8M (lowest line, extending below the axis on the right). The lines are parallel, indicating constant input prices, and are progressively closer to the origin as cost decreases. A blue dot is located at approximately (5,000, 46,000) on the isoquant curve, where it intersects or is near the TC = $2M isocost line, representing a higher-cost but feasible input combination. A red dot located at approximately (40,000, 10,000) on the isoquant curve, where it is tangent to the TC = $1.2M isocost line, representing the cost-minimizing input combination, the point where the isoquant is tangent to the lowest reachable isocost line, satisfying the condition that the marginal rate of technical substitution equals the input price ratio.<\/span><\/p>\r\n

Figure 7.2<\/a>: Line chart titled \"Market Yield on U.S. Treasury Securities at 1-Year Constant Maturity, Quoted on an Investment Basis.\" The vertical axis shows percent ranging from 0.0 to 18.0, and the horizontal axis shows dates at approximately decade intervals from April 1953 to approximately 2022. Vertical gray shaded bands mark periods of U.S. economic recessions throughout the chart. The line traces the history of the 1-year Treasury yield across seven decades, revealing several distinct phases: 1953\u2013early 1960s shows yields begin at approximately 1\u20132% and rise gradually with moderate cyclical fluctuations to approximately 4\u20135% by the early 1960s. 1960s\u20131981 (long secular rise) shows yields rise persistently through the 1960s and 1970s. The rate climbs from approximately 3% in the early 1960s to a dramatic peak of approximately 16.5% in mid-1981, the highest value in the entire series. 1981\u20132008 (long secular decline) shows following the 1981 peak, yields decline persistently and substantially over nearly three decades, falling from 16.5% to approximately 1\u20132% by the mid-2000s, interrupted by cyclical rises during expansion periods (notably approximately 9.5% in the late 1980s and approximately 6.5% around 2000). 2008\u20132021 (near-zero era) shows yields collapse to near zero following the Great Recession and remain at historically unprecedented low levels, at or near 0%, for most of the period from 2009 through 2021, reflecting the Federal Reserve's zero interest rate policy (ZIRP) and quantitative easing programs. 2021\u20132022 (rapid rise) shows yields begin rising sharply from near zero to approximately 4\u20135% by late 2022.<\/span><\/p>\r\n

Figure 7.3<\/a>:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
date<\/th>\r\n0.25<\/th>\r\n0.5<\/th>\r\n1<\/th>\r\n2<\/th>\r\n3<\/th>\r\n5<\/th>\r\n7<\/th>\r\n10<\/th>\r\n20<\/th>\r\n30<\/th>\r\n<\/tr>\r\n
09\/01\/1981<\/td>\r\n15.61<\/td>\r\n16.36<\/td>\r\n16.52<\/td>\r\n16.46<\/td>\r\n16.22<\/td>\r\n15.93<\/td>\r\n15.65<\/td>\r\n15.32<\/td>\r\n15.07<\/td>\r\n14.67<\/td>\r\n<\/tr>\r\n
10\/01\/1981<\/td>\r\n14.23<\/td>\r\n15.06<\/td>\r\n15.28<\/td>\r\n15.54<\/td>\r\n15.50<\/td>\r\n15.41<\/td>\r\n15.33<\/td>\r\n15.15<\/td>\r\n15.13<\/td>\r\n14.68<\/td>\r\n<\/tr>\r\n
11\/01\/1981<\/td>\r\n11.35<\/td>\r\n12.12<\/td>\r\n12.41<\/td>\r\n12.88<\/td>\r\n13.11<\/td>\r\n13.38<\/td>\r\n13.42<\/td>\r\n13.39<\/td>\r\n13.56<\/td>\r\n13.35<\/td>\r\n<\/tr>\r\n
12\/01\/1981<\/td>\r\n11.32<\/td>\r\n12.44<\/td>\r\n12.85<\/td>\r\n13.29<\/td>\r\n13.66<\/td>\r\n13.60<\/td>\r\n13.62<\/td>\r\n13.72<\/td>\r\n13.73<\/td>\r\n13.45<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n

Figure 7.4<\/a>: Side-by-side pair of line charts, both titled \"Yield Curve,\" displaying the relationship between bond maturity (in years) and yield (in percent) at two different points in time, illustrating contrasting yield curve shapes. The left chart shows an inverted yield curve. The vertical axis shows yield (%) ranging from 14.50 to 17.00, and the horizontal axis shows maturity in years from 0 to 35. The curve begins at approximately 15.6% at very short maturities, rises sharply to a peak of approximately 16.5% at around 1\u20132 years maturity, then declines steeply and persistently across longer maturities, falling to approximately 15.8% at 5 years, approximately 15.35% at 10 years, approximately 15.1% at 20 years, and approximately 14.75% at 30 years. This downward-sloping shape after the short-term peak is characteristic of an inverted yield curve, where short-term interest rates exceed long-term rates. The right chart is the normal yield curve. The vertical axis shows yield (%) ranging from 0.00 to 8.00, and the horizontal axis shows maturity in years from 0 to 35. The curve begins at approximately 5.8\u20136.0% at very short maturities and rises gradually and smoothly to approximately 6.6\u20136.8% at intermediate maturities of 5\u201310 years, then flattens and remains approximately level through 20 and 30 year maturities at approximately 6.7\u20136.8%. This upward-sloping then flattening shape is characteristic of a normal yield curve, where longer-term bonds carry modestly higher yields than short-term bonds, reflecting typical term premium and inflation expectations under normal economic conditions.<\/span><\/p>\r\n

Figure 7.6<\/a>: Three-dimensional surface chart titled \"Yield Curve over Time\" displaying the evolution of the U.S. Treasury yield curve across all maturities from 1981 to 2022. The three axes are: Yield percentage (vertical axis, ranging from 0.00 to 20.00), Date (diagonal left axis, running from 09\/01\/1981 at the back left to 11\/01\/2022 at the front right), and Maturity in years (diagonal right axis, with values visible at 20 and 0.25 years marked at the front right edge). The three-dimensional surface is rendered as a series of stacked yield curve cross-sections over time, with each thin slice representing the yield curve shape at a specific date. The color of the surface transitions from gray at the back (earliest dates in the early 1980s) through orange in the middle period (roughly 1990s\u2013early 2000s) to predominantly blue in the most recent period (2010s\u20132022), with the color variation providing additional visual differentiation across time. Early 1980s (back of the chart, gray) shows the surface begins at very high yield levels of approximately 14\u201316%, consistent with the peak interest rate environment of 1981\u20131982, with the yield curve appearing relatively flat or inverted at these extreme levels. 1980s\u20131990s (orange section) shows yields decline substantially from their early 1980s peaks, with the surface descending steeply. The yield curve takes on a more normal upward-sloping shape as short-term rates fall faster than long-term rates. 2000s\u20132010s (transition to blue) shows yields continue declining toward historically low levels, with the surface flattening considerably in the vertical dimension as rates approach zero. 2010s\u20132021 (blue, near-zero era) shows the surface compresses toward near-zero yield levels across most maturities, reflecting the extended zero interest rate policy period. The yield curve shape during this period shows a very flat profile at low absolute levels. 2022 (front of chart, orange reappearance) shows a sharp upward spike is visible at the most recent dates, reflecting the rapid Federal Reserve rate hiking cycle of 2022, with short-term yields rising dramatically and the curve potentially inverting again.<\/span><\/p>\r\n

Figure 7.7<\/a>: Three-dimensional surface chart displaying the evolution of the U.S. Treasury yield curve over time, rendered with a smooth continuous surface and a vibrant color gradient. The three axes are: Yield (%) on the vertical axis ranging from 0 to 16, Date on the left horizontal axis running from approximately 1985 to 2022, and Maturity on the right horizontal axis showing values of 0.5, 2, 5, 10, and 30 years. The surface is colored using a heat-map gradient transitioning from bright yellow at the highest yield levels at the back-upper portion of the surface (early 1980s, long maturities), through orange and red at intermediate yield levels, to deep purple and violet at the lowest yield levels toward the front of the chart (recent years, near-zero rates). This color gradient provides an intuitive visual indication of yield levels independent of the vertical axis. A tooltip annotation is visible on the surface reading: x: 30, y: Jul 1984, z: 13.21, indicating that the 30-year Treasury yield in July 1984 was 13.21%, one of the data points on the high-yield plateau visible at the back of the chart. 1984\u2013early 1990s (yellow-orange region) shows the surface begins at high yield levels of approximately 13\u201316% for long maturities in the mid-1980s, with the entire surface elevated. The yield curve during this period shows a relatively flat or moderately normal shape across maturities. 1990s\u20132000s (orange-red-pink region) shows the surface descends steadily as yields decline across all maturities, with the characteristic ripple pattern visible across the maturity dimension indicating recurring periods of yield curve steepening and flattening as the business cycle progresses. 2008\u20132021 (deep purple region) shows the surface compresses dramatically toward near-zero yield levels, particularly at short maturities, while long maturity yields remain slightly higher, creating visible ridges in the surface representing the persistent but low positive slope of the yield curve during the zero interest rate policy era. 2022 (front right, slight upward spike) shows a modest rise in yields is visible at the most recent dates, consistent with the beginning of the Federal Reserve rate hiking cycle.<\/span><\/p>\r\n

Figure 7.8<\/a>: Line chart titled \"10-Year Treasury Constant Maturity Minus 2-Year Treasury Constant Maturity.\" The vertical axis shows percent ranging from -3.0 to 4.0, and the horizontal axis shows dates at approximately seven-year intervals from June 1976 to September 2023. Vertical gray shaded bands mark periods of U.S. economic recessions. The chart displays the Treasury yield curve spread, the difference between the 10-year and 2-year Treasury yields, which is one of the most widely watched recession indicators. When the spread is positive, the yield curve is normal (upward sloping); when negative, the yield curve is inverted, historically a reliable predictor of recession. 1976\u20131980 shows the spread begins at approximately +1.5%, dips briefly into negative territory around 1978\u20131979 (yield curve inversion), then recovers, preceding the recession of the early 1980s. 1980\u20131982 shows a dramatic inversion to approximately -2.2% to -2.5% occurs, the deepest inversion in the entire series, coinciding with the Fed's aggressive tightening under Volcker and preceding the severe 1981\u20131982 recession. 1983\u20131989 shows the spread recovers sharply to approximately +2.5% as the yield curve normalizes, then gradually flattens back toward zero by the late 1980s, with a brief inversion preceding the 1990\u20131991 recession. 1990\u20132000 shows the spread cycles from near zero back up to approximately +2.5% in the early 1990s recovery, then declines again toward zero and briefly negative around 1998\u20132000, preceding the 2001 recession. 2001\u20132007 shows the spread recovers to approximately +2.5%, then declines steadily toward zero and briefly inverts around 2006\u20132007, preceding the Great Recession of 2008\u20132009. 2008\u20132015 shows the spread widens dramatically to nearly +2.8% as short-term rates are cut to zero while long-term rates remain higher, then gradually narrows through 2015\u20132018. 2018\u20132019 shows the spread approaches zero and briefly inverts slightly, signaling potential recession risk before the COVID-19 pandemic. 2022\u20132023 shows the spread inverts sharply to approximately -1.0%, one of the more significant inversions in the series, reflecting the Federal Reserve's rapid rate hiking cycle pushing short-term rates above long-term rates, consistent with elevated recession risk warnings.<\/span><\/p>\r\n

Figure 8.2<\/a>: Line chart titled \"Real Gross Domestic Product\" showing the quarterly compound annual percent change in U.S. real GDP from April 1947 to approximately 2022. The vertical axis shows compound annual percent change ranging from -40.0 to 40.0, and the horizontal axis shows dates at approximately decade intervals. Vertical gray shaded bands mark periods of U.S. economic recessions throughout the chart. The chart displays quarterly real GDP growth rates, which fluctuate considerably around a generally positive mean throughout the entire postwar period. 1947\u20131960s (high volatility era) show growth rates are highly volatile in the early postwar period, with frequent swings between approximately -10% and +17%. The amplitude of quarterly fluctuations is notably larger than in later decades. 1970s\u20131980s (stagflation and recession era) show growth remains volatile with several notable negative quarters coinciding with the oil shocks and recessions of 1973\u20131975 and 1981\u20131982, with troughs reaching approximately -10% and -8% respectively. A notable spike to approximately +16% occurs around 1978. 1990s\u20132000s (Great Moderation) shows quarterly GDP growth becomes notably less volatile, fluctuating within a narrower band of approximately -3% to +8%, reflecting the era of reduced macroeconomic volatility known as the Great Moderation. The 2008\u20132009 Great Recession produces a trough of approximately -8% to -9%. 2020\u20132021 (COVID-19 shock) shows that by far the most dramatic feature of the entire chart occurs at the far right, where the COVID-19 pandemic produces an unprecedented collapse in Q2 2020 to approximately -29% to -31% annualized, the largest quarterly decline in the entire postwar series by a wide margin, followed immediately by an equally unprecedented rebound to approximately +33% to +35% in Q3 2020 as the economy partially reopened. Growth then normalizes to more typical levels through 2021\u20132022.<\/span><\/p>\r\n

Figure 8.3<\/a>: Line chart displaying the quarterly compound annual percent change for three components of U.S. GDP from April 1947 to approximately 2022. The vertical axis shows percent change ranging from -100.0 to 150.0, and the horizontal axis shows dates at approximately decade intervals. A legend identifies three series. PCECC96 (blue line) represents real Personal Consumption Expenditures (PCE), the largest component of GDP. The blue line tracks closely along the zero line throughout the entire period with relatively modest fluctuations, generally ranging between approximately -5% and +10%, reflecting the relative stability of consumer spending compared to other GDP components. The line is nearly indistinguishable from the baseline for much of the chart due to its small amplitude relative to the other two series. GCEC1 (orange line) represents real Government Consumption Expenditures and Gross Investment. The orange line shows moderate volatility, generally fluctuating between approximately -10% and +65%, with the largest spikes occurring in the late 1940s and early 1950s and more modest fluctuations thereafter. Government spending volatility diminishes considerably from the 1960s onward. GPDIC1 (green line) represents real Gross Private Domestic Investment. By far the most volatile of the three series, the green line dominates the visual scale of the chart with dramatic swings throughout the entire period. Early postwar spikes reach approximately +100\u2013135% and drops to approximately -100% in the late 1940s. Throughout the series, private investment exhibits much larger cyclical swings than consumption or government spending, consistent with the well-established economic principle that investment is the most volatile component of GDP, amplifying business cycle fluctuations. The COVID-19 period produces another large spike and trough visible at the far right.<\/span><\/p>\r\n

Figure 8.4<\/a>:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Country<\/th>\r\nFRED code<\/th>\r\nCountry<\/th>\r\nFRED code<\/th>\r\nCountry<\/th>\r\nFRED code<\/th>\r\nCountry<\/th>\r\nFRED code<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Australia<\/td>\r\nAU<\/td>\r\nFrance<\/strong><\/td>\r\nFR<\/strong><\/td>\r\nJapan<\/strong><\/td>\r\nJP<\/strong><\/td>\r\nSlovak Republic<\/td>\r\nSK<\/td>\r\n<\/tr>\r\n
Austria<\/td>\r\nAT<\/td>\r\nGermany<\/strong><\/td>\r\nDE<\/strong><\/td>\r\nKorea<\/td>\r\nKR<\/td>\r\nSlovenia<\/td>\r\nSI<\/td>\r\n<\/tr>\r\n
Belgium<\/td>\r\nBE<\/td>\r\nGreece<\/td>\r\nGR<\/td>\r\nLuxembourg<\/td>\r\nLU<\/td>\r\nSouth Africa<\/td>\r\nZA<\/td>\r\n<\/tr>\r\n
Brazil<\/td>\r\nBR<\/td>\r\nHungary<\/td>\r\nHU<\/td>\r\nMexico<\/td>\r\nMX<\/td>\r\nSpain<\/td>\r\nES<\/td>\r\n<\/tr>\r\n
Canada<\/strong><\/td>\r\nCA<\/strong><\/td>\r\nIceland<\/td>\r\nIS<\/td>\r\nNetherlands<\/td>\r\nNL<\/td>\r\nSweden<\/td>\r\nSE<\/td>\r\n<\/tr>\r\n
Chile<\/td>\r\nCL<\/td>\r\nIdia<\/td>\r\nIN<\/td>\r\nNewZealand<\/td>\r\nNZ<\/td>\r\nSwitzerland<\/td>\r\nCH<\/td>\r\n<\/tr>\r\n
Czech Republic<\/td>\r\nCZ<\/td>\r\nIndonesia<\/td>\r\nID<\/td>\r\nNorway<\/td>\r\nNO<\/td>\r\nTurkey<\/td>\r\nTR<\/td>\r\n<\/tr>\r\n
Denmark<\/td>\r\nDK<\/td>\r\nIreland<\/td>\r\nIE<\/td>\r\nPoland<\/td>\r\nPL<\/td>\r\nUnited Kingdom<\/strong><\/td>\r\nGB<\/strong><\/td>\r\n<\/tr>\r\n
Estonia<\/td>\r\nEE<\/td>\r\nIsrael<\/td>\r\nIL<\/td>\r\nPortugal<\/td>\r\nPT<\/td>\r\nUnited States<\/strong><\/td>\r\nUS<\/strong><\/td>\r\n<\/tr>\r\n
Finland<\/td>\r\nFI<\/td>\r\nItaly<\/strong><\/td>\r\nIT<\/strong><\/td>\r\nRussia<\/td>\r\nRU<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

Figure 8.5<\/a>: Screenshot of the bottom rows of a Microsoft Excel spreadsheet, showing rows 313 through 315 and columns A through H, displaying the final data row, standard deviation formulas, and calculated results for four data series. Row 313 (final data row) contains the last data entries for four paired date-value columns. Column A: 07\/01\/2023, Column B: 4.9, Column C: 07\/01\/2023, Column D: 4.0, Column E: 07\/01\/2023, Column F: 8.4, Column G: 07\/01\/2023, Column H: 4.6. All four series share the same end date of July 1, 2023, suggesting the data runs from some earlier start date through mid-2023, with row 313 being the last observation (implying approximately 308 data points per series starting from row 6). Row 314 (formula row) contains Excel STDEV.P (population standard deviation) formulas for each value column: Column B: =STDEV.P(B6:B313), Column D: =STDEV.P(D6:D313), Column F: =STDEV.P(F6:F313), Column H: =STDEV.P(H6:H313). These formulas calculate the population standard deviation across the full range of each data series. Row 315 (results row) displays the calculated standard deviation values with a \"SD -->\" label in Column A: Column B: 4.6, Column D: 4.4, Column F: 20.9, Column H: 7.4.<\/span><\/p>\r\n

Figure 10.1<\/a>: Population pyramid chart labeled \"Year = 24\" displaying the age-sex distribution of a population at year 24 of a simulation or projection. The horizontal axis shows population counts ranging from 30,000 on the left to 30,000 on the right, with zero at the center. Blue bars extend to the left representing one sex (typically male) and red bars extend to the right representing the other sex (typically female). The chart shows two distinct population clusters separated by a gap. The upper cluster (larger, rectangular shape) is a broad, roughly rectangular block of blue (left) and red (right) bars centered at approximately 10,000\u201312,000 individuals per bar, with relatively uniform width across several age groups. The rectangular shape suggests a cohort of similar size moving through the age structure, consistent with a baby boom or large generational cohort. The lower cluster (smaller, irregular hourglass or star shape) is a smaller, more irregular cluster below the upper group, with a distinctive pinched or hourglass shape, wider at the top and bottom of this cluster and narrower in the middle. This unusual shape suggests a cohort with uneven age distribution, possibly reflecting a baby bust, mortality event, or other demographic disruption between the two clusters.<\/span><\/p>\r\n

Figure 10.2<\/a>: Population pyramid chart labeled \"Year = 100\" displaying the age-sex distribution of a population at year 100 of a simulation or projection, serving as a companion to the Year = 24 chart. The horizontal axis shows population counts ranging from 30,000 on the left to 30,000 on the right, with zero at the center. Blue bars extend to the left representing one sex (typically male) and red bars extend to the right representing the other sex (typically female). By year 100 the population pyramid has evolved dramatically from the fragmented, two-cluster structure visible at year 24 into a single continuous, broadly triangular or roughly Christmas-tree-shaped distribution spanning the full vertical height of the chart. There are a few key features of the year 100 pyramid. The base (youngest age groups) are where the widest bars appear near the bottom, extending to approximately 25,000\u201328,000 on each side, indicating a large young population. Middle age groups show the pyramid narrows gradually and relatively smoothly moving upward through middle age cohorts, with modest irregularities in the mid-section, including a slight indentation or notch visible on both sides at approximately one-third of the way up. The upper age groups (oldest) shows the pyramid tapers to near zero at the top, reflecting normal mortality attrition among the oldest age groups. The overall shape shows the broadly triangular shape with a wide base and narrowing apex is characteristic of a growing population with relatively high birth rates and normal age-specific mortality. The two-sided symmetry between blue and red is approximate, consistent with roughly equal sex ratios at birth and similar survival patterns.<\/span><\/p>\r\n

Figure 10.5<\/a>: Line chart titled \"United States Dependency Ratio\" showing the number of dependents per working-age adult from 1980 to 2060, with historical data and projections. The vertical axis shows dependents per working-age adult ranging from 0 to 0.70, and the horizontal axis shows years from 1970 to 2070. Two line segments are plotted in different colors to distinguish historical data from projections. The blue line (historical data, approximately 1980\u20132024) shows the dependency ratio begins at approximately 0.51 in 1980, remains relatively flat through the mid-1980s, rises modestly to approximately 0.53 around 1990, then declines gradually through the 1990s and 2000s to a trough of approximately 0.49 around 2010, reflecting the working-age Baby Boomer generation at peak employment. The ratio then rises from approximately 2010 onward, accelerating through the 2010s and early 2020s as Baby Boomers begin retiring, reaching approximately 0.57\u20130.58 by approximately 2023\u20132024. The orange line (projected data, approximately 2024\u20132060) shows the projection continues the upward trend steeply from the historical endpoint, rising sharply to approximately 0.62\u20130.63 by 2030, then leveling off and remaining relatively flat between approximately 0.62 and 0.65 through the 2030s and 2040s, before rising slightly again to approximately 0.66 by 2060.<\/span><\/p>\r\n

Figure 10.6<\/a>: Line chart titled \"United States Old-age Dependency Ratio\" showing the number of elderly dependents per working-age adult from 1980 to 2060, with historical data and projections. The vertical axis shows dependents per working-age adult ranging from 0 to 0.45, and the horizontal axis shows years from 1970 to 2070. Two line segments are plotted in different colors to distinguish historical data from projections. The blue line (historical data, approximately 1980\u20132024) shows the old-age dependency ratio begins at approximately 0.17 in 1980 and remains relatively flat through the 1980s at approximately 0.17\u20130.19, reflecting the relatively small pre-Baby Boomer elderly population during this period. The ratio stays nearly flat through the 1990s at approximately 0.19\u20130.20, then begins a gradual rise through the 2000s as early retirees enter the elderly population. From approximately 2010 onward the rate of increase accelerates markedly, rising from approximately 0.19 in 2010 to approximately 0.25 by 2020 and approximately 0.29\u20130.30 by approximately 2024, reflecting the accelerating retirement of the large Baby Boomer generation. The orange line (projected data, approximately 2024\u20132060) shows the projection continues the steep upward trajectory, rising sharply to approximately 0.35 by 2030, then leveling off and plateauing at approximately 0.35\u20130.36 through the mid-2030s to mid-2040s as the Baby Boomer retirement wave completes. The ratio then resumes a gradual upward trend, reaching approximately 0.39\u20130.40 by 2060, reflecting ongoing increases in longevity and the aging of subsequent generations.<\/span><\/p>","rendered":"

Figure 2.6<\/a>:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Distance Run<\/th>\nDistance Swim<\/th>\nTime to Victim<\/th>\n<\/tr>\n<\/thead>\n
0<\/td>\n141.4213562<\/td>\n70.71067812<\/td>\n<\/tr>\n
5<\/td>\n137.9311422<\/td>\n69.96557112<\/td>\n<\/tr>\n
10<\/td>\n134.5362405<\/td>\n69.26812024<\/td>\n<\/tr>\n
15<\/td>\n131.2440475<\/td>\n68.62202374<\/td>\n<\/tr>\n
20<\/td>\n128.0624847<\/td>\n68.03124237<\/td>\n<\/tr>\n
25<\/td>\n125<\/td>\n67.5<\/td>\n<\/tr>\n
30<\/td>\n122.0655562<\/td>\n67.03277808<\/td>\n<\/tr>\n
35<\/td>\n119.2686044<\/td>\n66.63430221<\/td>\n<\/tr>\n
40<\/td>\n116.6190379<\/td>\n66.30951895<\/td>\n<\/tr>\n
45<\/td>\n114.1271221<\/td>\n66.06356105<\/td>\n<\/tr>\n
50<\/td>\n111.8033989<\/td>\n65.90169944<\/td>\n<\/tr>\n
55<\/td>\n109.658561<\/td>\n65.8292805<\/td>\n<\/tr>\n
60<\/td>\n107.7032961<\/td>\n65.85164807<\/td>\n<\/tr>\n
65<\/td>\n105.9481005<\/td>\n65.97405025<\/td>\n<\/tr>\n
70<\/td>\n104.4030651<\/td>\n66.20153254<\/td>\n<\/tr>\n
75<\/td>\n103.0776406<\/td>\n66.53882032<\/td>\n<\/tr>\n
80<\/td>\n101.9802903<\/td>\n66.99019514<\/td>\n<\/tr>\n
85<\/td>\n101.1187421<\/td>\n67.55937104<\/td>\n<\/tr>\n
90<\/td>\n100,4987562<\/td>\n68.24937811<\/td>\n<\/tr>\n
95<\/td>\n100.124922<\/td>\n69.06246099<\/td>\n<\/tr>\n
100<\/td>\n100<\/td>\n70<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Figure 2.9<\/a>: Scatter plot graph titled with axis labels “Time to Victim (seconds)” on the vertical axis (ranging from 55 to 75 seconds) and “Distance Run (meters)” on the horizontal axis (ranging from 0 to 120 meters). The data is plotted as blue dots forming a smooth U-shaped curve, with two orange highlighted data points labeled “Initial” and “New.” The curve begins at approximately 71 seconds at 0 meters run, decreases steadily to a minimum of approximately 59\u201360 seconds around 55\u201360 meters, then rises again back toward 70 seconds at 100 meters. This U-shape illustrates that there is an optimal distance to run along the shore before entering the water to minimize total rescue time, consistent with an optimization problem involving two different travel speeds (running vs. swimming). The “Initial” orange data point is marked at approximately 55 meters run and 66 seconds, indicating a starting guess or pre-optimization estimate. The “New” orange data point is marked at approximately 80 meters run and 59 seconds, indicating the improved or solver-optimized solution that minimizes time to reach the victim.<\/p>\n

Figure 2.10<\/a>: The diagram shows a horizontal baseline with several labeled reference points and a large curved arc above it representing the objective function. The horizontal axis contains four key labeled positions from left to right: “Initial value” (the starting point of the solver), a series of intermediate black dots representing successive iterations of the solver’s path, “Exact answer” (the true mathematical optimum, marked with a dashed vertical line), and “Solver’s answer” (the solver’s final computed result, marked with a dashed vertical line slightly to the right of the exact answer). A large smooth arc labeled “Objective function” spans the full width of the diagram, peaking above the exact answer location, representing the function being maximized or minimized. A smaller nested arc labeled “Solver’s path” illustrates the trajectory of the solver’s iterative search process, showing how the solver takes progressively smaller steps, depicted as a series of diminishing arcs and dots along the baseline, as it homes in on the optimum. The solver’s final answer lands slightly past the exact answer, illustrating that numerical solvers approximate but may not land precisely on the true optimum.<\/p>\n

Figure 2.11<\/a>: Screenshot of a Microsoft Excel spreadsheet with an embedded scatter plot chart and an open Solver Parameters dialog box, illustrating a profit maximization problem. Column A is labeled “x” and contains values from 0 to 5 in increments of 0.5. Column B is labeled “profits” and contains corresponding dollar values: $0.00 (x=0), $1.75 (x=0.5), $3.00 (x=1), $3.75 (x=1.5), $4.00 (x=2), $3.75 (x=2.5), $3.00 (x=3), $1.75 (x=3.5), $0.00 (x=4), -$2.25 (x=4.5), -$5.00 (x=5). The profit values form an inverted U-shape, peaking at x=2. Rows 17 and 18 show the solver’s current variable value of approximately 1.999999995 for x and 4 for profits, representing the solver’s optimized result. Scatter plot chart: Titled “profits,” the chart plots profits (vertical axis, ranging from -$6.00 to $5.00) against x units sold (horizontal axis, ranging from 0 to 6). The data points form a clear downward-opening parabolic curve, confirming a maximum profit around x=2. Solver Parameters dialog box: The objective cell is set to $A$18 (profits), the optimization direction is set to Max (selected), and the variable cell to be changed is $A$17 (the x value). The dialog is partially visible, cut off below the “By Changing Variable Cells” field.<\/p>\n

Figure 3.3<\/a>: Screenshot of a Monte Carlo simulation output page from an Excel add-in, consisting of two components: a summary statistics table and a histogram. Summary Statistics table (top) is a bordered table with two columns labeled “Summary Statistics” (blue header) and “Notes” (red header). The Notes column is empty. The statistics reported are: Average = 89.955, SD (standard deviation) = 3.1523, Max = 98.000, Min = 78.000. Histogram (bottom) is titled “Histogram of $C$1.” The chart displays the frequency distribution of simulation results as a bar chart with blue outlined bars. The horizontal axis runs from 78 to 98. The vertical axis is unlabeled. The distribution is approximately bell-shaped and roughly symmetrical, centered around the mean of approximately 90. The bars rise gradually from the left tail starting near 78, reach their peak height around values of 91\u201392, then taper off toward the right tail ending near 98.<\/p>\n

Figure 3.10<\/a>: Line chart titled “Infection Rate = 5.0%” (displayed in red at the top). The vertical axis is labeled “Number of Tests” ranging from 0 to 1000, and the horizontal axis is labeled “Group Size” ranging from 0 to 20. Four lines are plotted. Individual Tests (red horizontal line) show a flat line at 1000 tests, representing the baseline cost of testing every individual separately, regardless of group size. E[#Total Tests] (black curve) shows the expected total number of tests required under a group testing strategy. This curve forms a U-shape, starting high at small group sizes (approximately 600 at group size 2), decreasing to a minimum of approximately 420 at group size 5, then rising again as group size increases, reaching approximately 680 at group size 20. This U-shape identifies the optimal group size that minimizes total tests. E[#Pos Group Tests] (orange curve) shows the expected number of follow-up individual tests required for positive groups. This curve starts low (approximately 100 at group size 2) and increases steadily and smoothly as group size grows, reaching approximately 640 at group size 20, reflecting that larger groups are more likely to contain at least one infected individual. #Groups Tested (blue curve) show the number of initial group tests administered. This curve starts high (approximately 500 at group size 2) and decreases steadily as group size increases, reaching approximately 50 at group size 20, since fewer but larger groups are needed to cover the population. A vertical annotation on the horizontal axis marks GS=5* as the optimal group size that minimizes the total expected number of tests at the given 5.0% infection rate.<\/p>\n

Figure 3.11<\/a>: Line chart titled “Infection Rate = 1.0%” (displayed in red at the top) illustrating pooled\/group testing optimization at a lower infection rate than a companion 5.0% chart. The vertical axis is labeled “Number of Tests” ranging from 0 to 1000, and the horizontal axis is labeled “Group Size” ranging from 0 to 20. Four lines are plotted: Individual Tests (red horizontal line) shows a flat line at 1000 tests, representing the baseline cost of testing every individual separately. E[#Total Tests] (black curve) shows the expected total number of tests under a group testing strategy. The curve starts high at approximately 520 at group size 2, decreases steeply to a shallow minimum of approximately 180\u2013190 around group size 11, then rises very gradually toward approximately 240 at group size 20. The curve is much flatter near its minimum compared to the 5.0% infection rate chart, indicating less sensitivity to group size choice near the optimum. E[#Pos Group Tests] (orange curve) shows the expected number of follow-up individual tests for positive groups. This curve starts near 0 at small group sizes and increases slowly and steadily, reaching approximately 190 at group size 20, reflecting that at a 1.0% infection rate, groups are much less likely to test positive. #Groups Tested (blue curve) shows the number of initial group screens administered. This curve starts high at approximately 500 at group size 2 and decreases smoothly, reaching approximately 50 at group size 20. A dashed vertical line and annotation mark GS=11* as the optimal group size minimizing total expected tests at the 1.0% infection rate.<\/p>\n

Figure 3.12<\/a>: Monte Carlo simulation output page consisting of a summary statistics table and a histogram. Summary Statistics table (top) is a bordered table with two columns labeled “Summary Statistics” (blue header) and “Notes” (red header). The Notes column contains the annotation “n = 5” in red, indicating the sample size used in the simulation. The statistics reported are: Average = 0.367, SD = 0.1411, Max = 1.052, Min = 0.200. Histogram (bottom) is titled “Histogram of $D$4.” The chart displays the frequency distribution of simulation results as a step-style bar chart with blue outlined bars. The horizontal axis ranges from approximately 0.2 to 1.0+, representing the range of simulated output values. The vertical axis shows frequency counts (unlabeled). The distribution is strongly right-skewed (positively skewed), with the tallest bars concentrated at the left end near the minimum value of 0.2, and a long, gradually tapering tail extending to the right toward values above 1.0. The mode is near 0.2 and frequency drops off steeply and then more gradually as values increase.<\/p>\n

Figure 3.13<\/a>: Monte Carlo simulation output page from an Excel add-in, comparing two simulation results side by side using a summary statistics table and an overlaid histogram. Summary Statistics table (top) has three sections. The left section (pink header) shows statistics for cell $D$4: Average = 0.367, SD = 0.1414, Max = 1.110, Min = 0.200. The center section (blue header) shows statistics for DGP!$F$4: Average = 0.361, SD = 0.1638, Max = 1.107, Min = 0.160. The right “Notes” column (red text) states: “n= 5 (pink) v 4 (blue) with 100,000 reps” and “n=4 is optimal.” Histogram (bottom) is titled “Histogram of $D$4 And DGP!F$4.” The chart overlays two step-style frequency distribution curves. The pink line represents $D$4 (n=5) and the blue line represents DGP!F$4 (n=4). The horizontal axis ranges from approximately 0.16 to above 0.96. Both distributions are strongly right-skewed, with peaks near the left end and long tails extending to the right. The pink curve (n=5) peaks slightly higher and is positioned slightly to the right of the blue curve (n=4), which starts at a lower minimum value of 0.16 compared to 0.20 for n=5. The two distributions are otherwise closely overlapping throughout most of their range. A legend in the lower right identifies the two series. The comparison illustrates the difference in the distribution of the optimization metric between the two candidate group sizes.<\/p>\n

Figure 3.14<\/a>: Combined spreadsheet table and line chart illustrating an optimal search theory model showing expected total cost as a function of the number of searches for a single search cost scenario. Spreadsheet table (left): header text states “Prices U[0,1], q=1 and c=0.04,” indicating prices are uniformly distributed between 0 and 1, quantity equals 1, and search cost c=0.04. The table has four columns: Number of Searches (1\u201310), Expected Value of Price, Search Cost, and Expected Value of Total Cost. Values begin at 1 search with expected price = 0.500, search cost = 0.04, and total cost = 0.540, then decline as searches increase. Row 4 is highlighted in blue bold italic, showing: 4 searches, expected price = 0.200, search cost = 0.16, total cost = 0.360, identifying n*=4 as the optimal number of searches. Values beyond row 4 show total cost rising again: row 5 total cost = 0.367, continuing upward to 0.491 at 10 searches. The line chart (right) is titled “Expected Value of Total Cost =f(search).” The vertical axis is labeled “expected value of total cost” ranging from 0.000 to 0.600, and the horizontal axis is labeled “n (number of searches)” ranging from 0 to 12. A single blue U-shaped curve is plotted, starting high at approximately 0.540 at n=1, declining steeply to a minimum at approximately 0.360, then rising gradually back toward 0.500 at n=10. A dashed vertical blue line marks the minimum at n=4*, identifying the optimal number of searches that minimizes total expected cost at this search cost level of c=0.04.<\/p>\n

Figure 3.15<\/a>: Combined spreadsheet table and line chart illustrating an optimal search theory model comparing expected total cost as a function of the number of searches under two different search cost scenarios. Spreadsheet table (left): Header text states “Prices U[0,1], q=1 and c=0.04,” indicating prices are uniformly distributed between 0 and 1, quantity equals 1, and search cost c=0.04. The table has four columns: Number of Searches (1\u201315), Expected Value of Price, Search Cost, and Expected Value of Total Cost. Key values include: at 1 search, expected price = 0.500, search cost = 0.01, total cost = 0.510; costs decline as searches increase to a minimum; row 9 is highlighted in red bold italic, showing: 9 searches, expected price = 0.100, search cost = 0.09, total cost = 0.190, identifying n*=9 as the optimal number of searches for c=0.04. Values beyond row 9 show total cost beginning to rise again. Line chart (right): Titled “Expected Value of Total Cost =f(search) for c=0.04 and c=0.01.” The vertical axis is labeled “expected value of total cost” ranging from 0.000 to 0.600, and the horizontal axis is labeled “n (number of searches)” ranging from 0 to 16. Two curves are plotted: a blue curve labeled “TC c=0.01” (lower search cost) and a red curve labeled “TC c=0.04” (higher search cost). Both curves are U-shaped, declining steeply at first then rising gradually. The blue curve sits higher overall due to lower per-search cost allowing more searches, with its minimum marked by a dashed blue vertical line at n=4*. The red curve sits lower overall and reaches its minimum at a dashed red vertical line at n=9*. The chart illustrates that lower search costs lead to fewer optimal searches while higher search costs lead to more, consistent with optimal search theory.<\/p>\n

Figure 4.2<\/a>: Line chart titled “Racing Sequences” comparing the growth trajectories of arithmetic and geometric sequences over time. The vertical axis shows dollar values ranging from $0 to $600,000,000, and the horizontal axis shows days ranging from 0 to 230. The Arithmetic sequence (blue line) is a straight, steadily increasing diagonal line that grows linearly from $0 at day 0 to approximately $200,000,000 by around day 200, illustrating constant additive growth at a fixed amount per day. The geometric sequence (orange dotted line) is a curve that remains nearly flat and close to zero for the majority of the time period, virtually indistinguishable from the baseline until approximately day 175\u2013180, at which point it begins to curve sharply upward in a classic exponential growth pattern. By approximately day 200 it overtakes the arithmetic sequence, and by approximately day 215 it reaches nearly $500,000,000, far exceeding the arithmetic sequence’s value. The dotted styling of the geometric line may indicate projected or extrapolated values beyond a certain point.<\/p>\n

Figure 4.4<\/a>: Line chart comparing average income or wealth in the United States and Japan from approximately 1955 to 2022, measured in thousands of euros at 2022 purchasing power parity (PPP). The vertical axis shows values in thousand euros (2022) ranging from 0 to 70k, and the horizontal axis shows years from approximately 1955 to 2022. Two lines are plotted. The US line (red) begins at approximately \u20ac24,000\u201325,000 in the mid-1950s and rises steadily with periodic fluctuations throughout the entire period. Growth accelerates notably from the 1980s onward, reaching approximately \u20ac55,000 by 2000 and peaking near \u20ac68,000\u201369,000 around 2021\u20132022, with a brief dip around 2008\u20132009 consistent with the global financial crisis. The Japan line (blue) begins at approximately \u20ac5,000 in the mid-1950s and grows rapidly through the 1970s and 1980s, reflecting Japan’s postwar economic miracle. Growth slows considerably after approximately 1990, consistent with Japan’s “Lost Decade,” with the line flattening and plateauing around \u20ac33,000\u201338,000 through the 1990s and 2000s, before modest growth resumes toward approximately \u20ac37,000 by 2022.<\/span><\/p>\n

Figure 4.6<\/a>: Line chart showing college wage premium as a percentage for different racial and ethnic groups in the United States from 2000 to approximately 2022. The vertical axis is labeled “Wage premium (%)” ranging from 60 to 130, and the horizontal axis shows years from 2000 to 2022. Five lines are plotted, each representing a different demographic group. The Asian line (green) is the highest line throughout the entire period, beginning at approximately 82\u201383% in 2000 and rising with considerable year-to-year volatility to a peak of approximately 125% around 2019\u20132020, before declining to approximately 112% by 2022. The overall upward trend is strong and the Asian wage premium is substantially higher than all other groups throughout the period. Black (blue line) begins at approximately 76% in 2000, fluctuates moderately, reaches a peak of approximately 90% around 2013, then gradually declines toward approximately 72\u201375% by 2022, ending close to the overall average. Overall (red line, thick) is the bold reference line representing the average wage premium across all groups. Begins at approximately 69% in 2000 and rises gradually to approximately 77\u201378% by the early 2020s, with modest year-to-year variation. White (yellow\/orange line) tracks closely below the overall average throughout the period, beginning at approximately 67\u201368% in 2000 and rising modestly to approximately 72\u201374% by 2022. Hispanic (dark red\/brown line) is the lowest line for most of the period, beginning at approximately 68% in 2000, fluctuating between approximately 62% and 79%, and ending at approximately 70% by 2022, slightly below the overall average.<\/span><\/p>\n

Figure 4.7<\/a>: Line chart titled “Hypothetical Projected HS versus College Income” comparing cumulative or annual income trajectories for high school (HS) and college graduates over a working lifetime. The vertical axis shows dollar values ranging from -$50,000 to $200,000, and the horizontal axis shows age ranging from 18 to 68. Two curves are plotted: College line (orange curve) begins below zero at age 18, dipping into negative territory (toward -$50,000) to represent the cost of college tuition and foregone income during the college years. The curve rises steeply from approximately age 22 onward, crossing into positive income territory and continuing to grow at a diminishing rate, reaching approximately $185,000 by age 68. HS line (blue curve) begins at approximately $35,000 at age 18, representing immediate entry into the workforce after high school. The curve rises gradually and at a diminishing rate throughout the career, reaching approximately $110,000 by age 68. Three labeled regions illustrate the key financial components of the college investment decision: Negative A (marked near age 18 on the horizontal axis) represents the direct cost of college tuition and expenses, shown as the negative dip in the orange curve below zero. Negative B (marked with a dashed vertical orange line near age 22\u201323) represents the opportunity cost of college, the foregone income a college student gives up compared to a high school graduate working during those years. +C (labeled in the large region between the two curves during the working years) represents the cumulative earnings premium that college graduates earn over high school graduates throughout their career, which offsets the earlier costs A and B. The chart is a standard educational illustration of the human capital investment model, demonstrating that college has upfront costs (-A and -B) but generates long-term income benefits (+C) that, if sufficiently large, justify the investment.<\/span><\/p>\n

Figure 4.8<\/a>: Scatter plot chart titled “HS and College Income Streams” comparing the annual income of high school (HS) and college graduates as discrete yearly data points across a working lifetime. The vertical axis shows dollar values ranging from -$30,000 to $80,000, and the horizontal axis shows age ranging from 18 to 68. Two data series are plotted as dotted lines of individual points. College (orange dots) show ages 18, 19, 20, and 21. The college data points appear at approximately -$20,000, representing net negative income during the college years due to tuition costs and foregone earnings. A dashed vertical orange line marks approximately age 22, indicating the transition from college to workforce entry. From age 22 onward, the orange dots form a flat horizontal line at exactly $70,000 per year, continuing consistently through age 65\u201368, representing a constant annual college graduate salary.
\nHS (blue dots) begin at age 18. The blue dots form a flat horizontal line at exactly $40,000 per year, continuing consistently through age 65\u201368, representing a constant annual high school graduate salary throughout the entire working career with no interruption.<\/span><\/p>\n

Figure 5.3<\/a>: Line chart titled “Unemployment Rate” showing the U.S. unemployment rate as a percentage from January 1952 to approximately 2022. The vertical axis shows percent ranging from 0.0 to 16.0, and the horizontal axis shows dates at approximately decade intervals from Jan-52 to Jul-13 and beyond. Vertical gray shaded bands mark periods of U.S. economic recessions throughout the chart. The unemployment rate begins at approximately 3% in early 1952 and fluctuates cyclically throughout the entire period, with recurring peaks during recessions followed by recoveries to lower levels. Notable peaks include: approximately 6% in the mid-1950s recession; approximately 7\u20137.5% during the early 1960s recession; approximately 9% during the mid-1970s recession following the oil shock; a dramatic peak of approximately 10.8% in late 1982\u20131983 during the severe double-dip recession; approximately 7.8% in the early 1990s recession; approximately 6.3% in the early 2000s following the dot-com bust; approximately 10% in 2009\u20132010 during the Great Recession; and a catastrophic spike to approximately 14.7% in April 2020 due to the COVID-19 pandemic shutdown, the highest value in the entire series, followed by an extremely rapid recovery back to approximately 3.5\u20134% by 2022.<\/span><\/p>\n

Figure 5.4<\/a>: Line chart titled “Labor Force Participation Rate” showing the percentage of the U.S. civilian noninstitutional population participating in the labor force from January 1952 to approximately 2022. The vertical axis shows percent ranging from 52.0 to 68.0, and the horizontal axis shows dates at approximately decade intervals. Vertical gray shaded bands mark periods of U.S. economic recessions. The chart reveals four distinct phases. 1952\u2013early 1960s (flat\/slightly declining) shows the participation rate hovers in a narrow band between approximately 58.5% and 60.0%, with modest cyclical fluctuations around recessions but no clear trend. Mid-1960s through 2000 (long sustained rise) shows that the rate rises steadily and substantially from approximately 58.5% in the mid-1960s to a peak of approximately 67.3% around 2000, an increase of nearly 9 percentage points over roughly 35 years. The ascent is nearly continuous, interrupted only briefly by recessions, and accelerates noticeably through the 1970s and 1980s. 2000\u20132015 (prolonged decline) shows that following the peak around 2000, the participation rate declines persistently from approximately 67.3% to approximately 62.4% by 2015. The decline accelerates during and after the Great Recession (2008\u20132009). 2015\u20132022 (stabilization then COVID shock) shows the rate stabilizes around 62.5\u201363.5% before plunging sharply to approximately 60.2% in April 2020 due to the COVID-19 pandemic, then recovering partially to approximately 62.5% by 2022, remaining well below pre-pandemic and historical peak levels.<\/span><\/p>\n

Figure 5.6<\/a>: Line chart titled “Male and Female Unemployment Rate” comparing unemployment rates for women and men aged 20 and over in the United States from January 1948 to approximately 2022. The vertical axis shows percent ranging from 0.0 to 18.0, and the horizontal axis shows dates at approximately decade intervals from Jan-48 to Jan-13 and beyond. Two lines are plotted: Female 20+ (orange line) represents the unemployment rate for adult women aged 20 and older. Male 20+ (green line) represents the unemployment rate for adult men aged 20 and older. From 1948\u2013late 1960s, the two lines track closely together with moderate cyclical fluctuations, both ranging between approximately 2.5% and 7\u20138%. Female unemployment is occasionally slightly higher than male during this period. From 1970s\u20131980s, both series rise substantially, reflecting the stagflation era and recessions. Female unemployment peaks at approximately 8.5% and male at approximately 6% during the mid-1970s recession. During the severe 1981\u20131982 recession, male unemployment surges dramatically to approximately 10% while female peaks slightly lower at approximately 9.5%, marking one of the most notable periods where male unemployment exceeds female. From 1990s\u20132000s, the two lines converge and track very closely together, both cycling between approximately 3.5% and 7\u20138% through successive recessions, with female unemployment generally slightly above or equal to male. From 2008\u20132010 (Great Recession), male unemployment rises more sharply than female, peaking at approximately 10.5% compared to approximately 8% for females. 2020 (COVID-19 pandemic) shows both series spike dramatically. Female unemployment surges to approximately 15.5% while male reaches approximately 11.5%, with female unemployment exceeding male by the largest margin in the entire series. Both recover rapidly to approximately 3.5% by 2022.<\/span><\/p>\n

Figure 5.7<\/a>: Line chart titled “Overall and Teenage Unemployment Rate” comparing the overall U.S. unemployment rate with the teenage unemployment rate (ages 16\u201319) from January 1948 to approximately 2022. The vertical axis shows percent ranging from 0.0 to 35.0, and the horizontal axis shows dates at approximately decade intervals from Jan-48 onward. Two lines are plotted: Overall (blue line) represents the total U.S. unemployment rate across all age groups, tracking closely with the series shown in companion charts. The line fluctuates cyclically between approximately 2.5% and 10.8%, with recession-era peaks visible throughout the period and a dramatic spike to approximately 14.7% in April 2020 during the COVID-19 pandemic, before rapidly recovering to approximately 3.5% by 2022. Teenage 16\u201319yrs (green line) represents the unemployment rate for teenagers aged 16\u201319. This line runs consistently and substantially above the overall rate throughout the entire 74-year period, typically at a level two to three times higher than the overall rate at any given time. Key observations include: the teenage rate begins at approximately 9\u201315% in the late 1940s and early 1950s; it rises with each recession, peaking at approximately 24% during the severe 1981\u20131982 recession; it reaches approximately 27% during the Great Recession of 2009\u20132010; and it spikes to approximately 32.5% during the COVID-19 pandemic in 2020 before recovering sharply to approximately 10\u201311% by 2022. <\/span><\/p>\n

Figure 5.8<\/a>: Line chart titled “Unemployment Rate by Race and Ethnicity” comparing U.S. unemployment rates across five racial and ethnic groups from 1948 to approximately 2022. The vertical axis shows percent ranging from 0.0 to 25.0, and the horizontal axis shows years from 1948 to 2028. A legend identifies five series. Black (gray line) shows consistently the highest unemployment rate throughout the entire period. Begins at approximately 7\u20138% in the late 1940s and fluctuates cyclically, reaching a dramatic peak of approximately 21% during the 1981\u20131982 recession \u2014 the highest value of any group in the series. Subsequent recession peaks occur at approximately 16\u201317% in the early 1990s and approximately 16% during the Great Recession of 2009\u20132010. The COVID-19 pandemic spike reaches approximately 15\u201316% before recovering to approximately 5\u20136% by 2022. Throughout the entire period, Black unemployment remains roughly double the White rate. Hispanic (blue line) shows data beginning in the mid-1970s. It consistently tracks above the overall and White rates but below the Black rate for most of the period. It peaks at approximately 15.5% during the 1981\u20131982 recession and approximately 13% during the Great Recession. The COVID-19 spike reaches approximately 18.5%, briefly exceeding the Black rate, before recovering to approximately 4\u20135% by 2022. Overall (black line) is the reference series representing all workers combined, tracking between the higher-unemployment minority groups and the lower White rate. Cyclical peaks reach approximately 10\u201311% during the 1981\u20131982 recession and approximately 10% during the Great Recession, with a COVID spike to approximately 14.7%. White (orange line) is consistently among the lowest unemployment rates throughout the period, tracking closely below the overall rate. It peaks at approximately 9\u201310% during the 1981\u20131982 recession and approximately 9% during the Great Recession, with a COVID spike to approximately 14% before rapid recovery to approximately 3.5%. Asian (green line) has data beginning in the early 2000s and is available only for the most recent portion of the chart. It tracks at or below the White and overall rates during normal periods, reaching approximately 3\u20134% at cyclical lows. It experiences a sharp COVID-19 spike to approximately 15% in 2020 before recovering rapidly to approximately 3% by 2022.<\/span><\/p>\n

Figure 5.10<\/a>: Line chart comparing average Not Seasonally Adjusted (NSA) and Seasonally Adjusted (SA) unemployment or economic data by month across a full calendar year. The vertical axis shows values ranging from 0 to 7, and the horizontal axis shows all twelve months from January through December. A “Month” dropdown filter button is visible at the bottom left, and toggle buttons for “Average of NSA” and “Average of SA” appear at the top left. Two lines are plotted: Average of NSA (blue line) represents the raw, not seasonally adjusted average values by month. The line shows clear seasonal variation, beginning high at approximately 6.3 in January, remaining elevated at approximately 6.4 in February, then declining through spring to a trough of approximately 5.5 in April\u2013May, rising modestly to approximately 6.0 in June, then declining again through summer and fall to a low of approximately 5.2 in November, before recovering slightly to approximately 5.5 in December. The blue line oscillates noticeably throughout the year, reflecting true seasonal patterns in the underlying data. Average of SA (orange line) represents the seasonally adjusted average values by month. The line is nearly flat throughout the entire year, hovering consistently between approximately 5.6 and 5.8 with minimal month-to-month variation. This flat profile illustrates the effect of seasonal adjustment, which removes predictable seasonal fluctuations to reveal the underlying trend.<\/span><\/p>\n

Figure 6.3<\/a>: Three-dimensional surface chart titled “Utility Maximization” displaying a utility function over two consumption goods. The three axes are: Utility (vertical axis, ranging from approximately -120 to 130), Brandies (horizontal axis along the left base, ranging from 0 to approximately 10), and Cigars (horizontal axis along the right base, ranging from approximately 3 to 18). The surface forms a smooth, dome-shaped hill or inverted paraboloid, rising from low or negative utility values at the edges and corners of the consumption space to a clear single peak or maximum in the interior of the surface. The peak appears to occur at intermediate values of both brandies and cigars, at a utility level of approximately 120\u2013130. Beyond the optimal combination in any direction utility declines, eventually falling into negative values at the extremes of the axes. The surface is rendered with a multi-colored gradient from blue and green at the lower utility regions through yellow, orange, and red toward the peak, providing a visual heat-map-like indication of utility levels across the consumption space.<\/span><\/p>\n

Figure 6.5<\/a>: Contour plot titled “Utility Maximization: Contour Plot” displaying a top-down view of the utility surface from the companion three-dimensional surface chart. The horizontal axis is labeled “Brandies” ranging from 0 to 10, and the vertical axis is labeled “Cigars” ranging from 0 to 20. The plot uses a multi-colored concentric ring pattern to represent utility levels across all combinations of the two goods, with each colored band representing a contour line of equal utility (an indifference curve). The color gradient transitions from warm colors (orange and red) at the outermost rings representing lower utility levels, through yellow and green in the intermediate rings, to cool blue and light blue at the innermost rings representing the highest utility levels, converging toward a single optimal point at the center of the innermost ring. A red dot marks the utility-maximizing consumption bundle, and an annotation box connected to it by a line reads: “B=3, C=10 yields U*=127,” identifying the optimal solution as 3 brandies and 10 cigars, which produces a maximum utility value of 127. The contour plot provides a two-dimensional representation of the same utility surface shown in the 3D chart, making it easier to identify the precise location of the utility maximum and visualize how utility changes as consumption of either good moves away from the optimal bundle in any direction.<\/span><\/p>\n

Figure 6.6<\/a>: Side-by-side pair of charts titled “Constrained Utility Maximization: 3D Surface” (left) and “Constrained Utility Maximization: Contour Plot” (right), extending the unconstrained utility maximization charts by adding a budget constraint. The left panel is three-dimensional surface chart identical in structure to the unconstrained utility surface, with axes for Utility (vertical, approximately -120 to 130), Brandies (horizontal base axis, 0 to approximately 10), and Cigars (horizontal base axis, approximately 3 to 18). The same dome-shaped utility surface is displayed with a multi-colored gradient. A red planar surface cuts through the utility dome diagonally, representing the budget constraint \u2014 a flat plane that intersects the utility surface and limits the feasible consumption combinations to those at or below the budget. The constrained optimum occurs at the point where the budget constraint plane is tangent to the highest reachable point on the utility surface. The right panel is a top-down contour plot with the same concentric multi-colored rings as the unconstrained version, with Brandies on the horizontal axis (0 to 10) and Cigars on the vertical axis (0 to 20). A red diagonal line represents the budget constraint, running from Cigars, 5 to Brandies, 5 of the feasible consumption space, indicating the trade-off between purchasing brandies and cigars given a fixed budget. A light blue dot marks the constrained utility-maximizing consumption bundle on the budget line, and an annotation box reads: “B=1, C=4 yields U*=79,” identifying the optimal constrained solution as 1 brandy and 4 cigars, yielding a maximum attainable utility of 79, substantially lower than the unconstrained maximum of 127, illustrating the utility cost of the budget constraint.<\/span><\/p>\n

Figure 6.7<\/a>: Wireframe contour plot titled “Constrained Utility Maximization: Wireframe Contour” displaying the same utility maximization problem as the companion filled contour plot, but rendered as a wireframe (outline-only) version without color fill. The horizontal axis is labeled “Brandies” ranging from 0 to 10, and the vertical axis is labeled “Cigars” ranging from 0 to 20. The wireframe consists of concentric oval\/elliptical contour lines drawn in varying colors (blue, orange, yellow, green, and red) representing indifference curves of equal utility at different levels. The contours are centered around the unconstrained utility maximum located at approximately Brandies=3, Cigars=10, with inner rings representing higher utility levels and outer rings representing lower utility levels. The elliptical shape of the contours indicates that the utility function values the two goods differently, with the contours being more elongated along the Cigars axis than the Brandies axis. A red diagonal straight line runs from approximately the point (0, 5) on the Cigars axis down to approximately (5.5, 0) on the Brandies axis, representing the budget constraint. The line indicates the maximum affordable combinations of brandies and cigars given the consumer’s budget and the prices of each good. The constrained optimum occurs where the budget constraint line is tangent to the highest reachable indifference curve, which based on the companion contour plot is at B*=1, C*=4 yielding U*=79.<\/span><\/p>\n

Figure 6.8<\/a>: Small diagram illustrating the tangency condition for constrained utility maximization, showing three indifference curves and a budget constraint line. Three blue curved lines arc from the upper left to the lower right, representing indifference curves at three distinct utility levels, labeled on the right side as: U>79 (uppermost curve, highest utility level), U=79* (middle curve, the optimal utility level), and U<79 (lowest curve, lowest utility level shown). A red diagonal straight line represents the budget constraint, running from upper left to lower right across all three curves. The red budget constraint line intersects the lowest indifference curve (U<79) at two points, crosses through the middle curve (U*=79) at exactly one point of tangency, and does not reach the highest indifference curve (U>79), which lies entirely beyond the budget constraint.<\/span><\/p>\n

Figure 6.10<\/a>: Line chart titled “B* = f(Total)” showing the relationship between total budget allowed and the optimal consumption of brandies in a constrained utility maximization model. The horizontal axis is labeled “Total Allowed” ranging from 0 to 12, representing the consumer’s total budget or spending limit. The vertical axis is labeled “Optimal Brandies Consumption” ranging from 0 to 2.5, representing the utility-maximizing quantity of brandies given each budget level. A single blue line with data point markers connects seven plotted points, beginning at approximately (5, 1.0) and rising steadily and nearly linearly to approximately (10, 2.25). The data points and approximate values are: Total=5 \u2192 B*\u22481.00; Total=6 \u2192 B*\u22481.25; Total=7 \u2192 B*\u22481.50; Total=8 \u2192 B*\u22481.75; Total=9 \u2192 B*\u22482.00; Total=10 \u2192 B*\u22482.25. No data points are plotted below a total budget of 5.<\/span><\/p>\n

Figure 6.11<\/a>: Horizontal number line diagram illustrating the concept of elasticity in economics, ranging from negative infinity (\u2212\u221e) on the left to positive infinity (+\u221e) on the right, with key values marked at \u22122, \u22121, 0, +1, and +2. The diagram is divided into three conceptual zones with labels above the number line. The left side (negative values) are labeled “Negative Inverse relationship,” with two downward-pointing arrows indicating that when the elasticity is negative, the two variables move in opposite directions (one rises while the other falls). A leftward arrow labeled “Responsiveness Increasing” runs from 0 toward \u2212\u221e, indicating that as the elasticity value becomes more negative, the responsiveness or sensitivity of the relationship increases. The center (zero) is labeled “No relationship,” with a downward arrow pointing to 0 on the number line, indicating that an elasticity of zero means the dependent variable does not respond at all to changes in the independent variable. The right side (positive values) is labeled “Positive Direct relationship,” with two upward-pointing arrows indicating that when the elasticity is positive, the two variables move in the same direction. A rightward arrow labeled “Responsiveness Increasing” runs from 0 toward +\u221e, indicating that as the elasticity value becomes more positive, responsiveness increases. Below the number line, five key elasticity benchmarks are labeled at their corresponding positions: Perfectly Elastic (at \u2212\u221e), Unit Elastic (at \u22121), Perfectly Inelastic (at 0), Unit Elastic (at +1), and Perfectly Elastic (at +\u221e).<\/span><\/p>\n

Figure 6.12<\/a>: Line chart titled “Sales of cigarettes per adult per day.” The subtitle notes that figures include manufactured cigarettes as well as estimated hand-rolled cigarettes per adult aged 15 and over per day. The vertical axis shows cigarette quantities from 0 to approximately 11 cigarettes, and the horizontal axis shows years from approximately 1900 to 2015. Two lines are plotted. United States (red line) begins near 0 cigarettes per adult per day around 1900 and rises steeply and steadily through the early twentieth century, reaching approximately 2 cigarettes by 1920. Growth continues through the 1930s and accelerates sharply in the 1940s and 1950s, peaking at approximately 10.5\u201311 cigarettes per adult per day around 1963\u20131965, the highest value in the entire chart. Following the peak, the US line declines persistently and steadily through the 1970s, 1980s, 1990s, and 2000s, reaching approximately 3 cigarettes per day by 2015. Japan (blue line) data begins in the early 1900s at low levels, tracking similarly to the US through the 1920s and 1930s at approximately 2 cigarettes. A sharp dip occurs around 1945\u20131946, dropping to approximately 1 cigarette. Japan’s consumption rises steeply through the 1950s, 1960s, and 1970s, peaking at approximately 9.5\u201310 cigarettes per adult per day around 1977\u20131980, roughly a decade after the US peak. Japan’s subsequent decline is slower and more gradual than the US, reaching approximately 4.5 cigarettes per day by 2015, remaining substantially higher than the US at the end of the series.<\/span><\/p>\n

Figure 6.14<\/a>: Line chart illustrating a cost minimization problem in microeconomics or operations management, showing an isoquant curve and three isocost lines. The vertical axis ranges from 0 to 100,000 and the horizontal axis ranges from 0 to 100,000, with both axes representing quantities of two inputs (unlabeled). Isoquant curve (thick dark blue curve) is a downward-sloping convex curve labeled “q=300.” The curve begins at approximately (1,000, 90,000) in the upper left and sweeps downward to near (100,000, 0) at the lower right, exhibiting the typical diminishing marginal rate of technical substitution shape. Three isocost lines (thin straight diagonal blue lines): Three parallel downward-sloping straight lines representing different total cost levels, labeled on the right side of the chart: TC = $2M (uppermost line, highest cost), TC = $1.2M (middle line), and TC = $0.8M (lowest line, extending below the axis on the right). The lines are parallel, indicating constant input prices, and are progressively closer to the origin as cost decreases. A blue dot is located at approximately (5,000, 46,000) on the isoquant curve, where it intersects or is near the TC = $2M isocost line, representing a higher-cost but feasible input combination. A red dot located at approximately (40,000, 10,000) on the isoquant curve, where it is tangent to the TC = $1.2M isocost line, representing the cost-minimizing input combination, the point where the isoquant is tangent to the lowest reachable isocost line, satisfying the condition that the marginal rate of technical substitution equals the input price ratio.<\/span><\/p>\n

Figure 7.2<\/a>: Line chart titled “Market Yield on U.S. Treasury Securities at 1-Year Constant Maturity, Quoted on an Investment Basis.” The vertical axis shows percent ranging from 0.0 to 18.0, and the horizontal axis shows dates at approximately decade intervals from April 1953 to approximately 2022. Vertical gray shaded bands mark periods of U.S. economic recessions throughout the chart. The line traces the history of the 1-year Treasury yield across seven decades, revealing several distinct phases: 1953\u2013early 1960s shows yields begin at approximately 1\u20132% and rise gradually with moderate cyclical fluctuations to approximately 4\u20135% by the early 1960s. 1960s\u20131981 (long secular rise) shows yields rise persistently through the 1960s and 1970s. The rate climbs from approximately 3% in the early 1960s to a dramatic peak of approximately 16.5% in mid-1981, the highest value in the entire series. 1981\u20132008 (long secular decline) shows following the 1981 peak, yields decline persistently and substantially over nearly three decades, falling from 16.5% to approximately 1\u20132% by the mid-2000s, interrupted by cyclical rises during expansion periods (notably approximately 9.5% in the late 1980s and approximately 6.5% around 2000). 2008\u20132021 (near-zero era) shows yields collapse to near zero following the Great Recession and remain at historically unprecedented low levels, at or near 0%, for most of the period from 2009 through 2021, reflecting the Federal Reserve’s zero interest rate policy (ZIRP) and quantitative easing programs. 2021\u20132022 (rapid rise) shows yields begin rising sharply from near zero to approximately 4\u20135% by late 2022.<\/span><\/p>\n

Figure 7.3<\/a>:<\/p>\n\n\n\n\n\n\n\n
date<\/th>\n0.25<\/th>\n0.5<\/th>\n1<\/th>\n2<\/th>\n3<\/th>\n5<\/th>\n7<\/th>\n10<\/th>\n20<\/th>\n30<\/th>\n<\/tr>\n
09\/01\/1981<\/td>\n15.61<\/td>\n16.36<\/td>\n16.52<\/td>\n16.46<\/td>\n16.22<\/td>\n15.93<\/td>\n15.65<\/td>\n15.32<\/td>\n15.07<\/td>\n14.67<\/td>\n<\/tr>\n
10\/01\/1981<\/td>\n14.23<\/td>\n15.06<\/td>\n15.28<\/td>\n15.54<\/td>\n15.50<\/td>\n15.41<\/td>\n15.33<\/td>\n15.15<\/td>\n15.13<\/td>\n14.68<\/td>\n<\/tr>\n
11\/01\/1981<\/td>\n11.35<\/td>\n12.12<\/td>\n12.41<\/td>\n12.88<\/td>\n13.11<\/td>\n13.38<\/td>\n13.42<\/td>\n13.39<\/td>\n13.56<\/td>\n13.35<\/td>\n<\/tr>\n
12\/01\/1981<\/td>\n11.32<\/td>\n12.44<\/td>\n12.85<\/td>\n13.29<\/td>\n13.66<\/td>\n13.60<\/td>\n13.62<\/td>\n13.72<\/td>\n13.73<\/td>\n13.45<\/td>\n<\/tr>\n<\/thead>\n<\/table>\n

Figure 7.4<\/a>: Side-by-side pair of line charts, both titled “Yield Curve,” displaying the relationship between bond maturity (in years) and yield (in percent) at two different points in time, illustrating contrasting yield curve shapes. The left chart shows an inverted yield curve. The vertical axis shows yield (%) ranging from 14.50 to 17.00, and the horizontal axis shows maturity in years from 0 to 35. The curve begins at approximately 15.6% at very short maturities, rises sharply to a peak of approximately 16.5% at around 1\u20132 years maturity, then declines steeply and persistently across longer maturities, falling to approximately 15.8% at 5 years, approximately 15.35% at 10 years, approximately 15.1% at 20 years, and approximately 14.75% at 30 years. This downward-sloping shape after the short-term peak is characteristic of an inverted yield curve, where short-term interest rates exceed long-term rates. The right chart is the normal yield curve. The vertical axis shows yield (%) ranging from 0.00 to 8.00, and the horizontal axis shows maturity in years from 0 to 35. The curve begins at approximately 5.8\u20136.0% at very short maturities and rises gradually and smoothly to approximately 6.6\u20136.8% at intermediate maturities of 5\u201310 years, then flattens and remains approximately level through 20 and 30 year maturities at approximately 6.7\u20136.8%. This upward-sloping then flattening shape is characteristic of a normal yield curve, where longer-term bonds carry modestly higher yields than short-term bonds, reflecting typical term premium and inflation expectations under normal economic conditions.<\/span><\/p>\n

Figure 7.6<\/a>: Three-dimensional surface chart titled “Yield Curve over Time” displaying the evolution of the U.S. Treasury yield curve across all maturities from 1981 to 2022. The three axes are: Yield percentage (vertical axis, ranging from 0.00 to 20.00), Date (diagonal left axis, running from 09\/01\/1981 at the back left to 11\/01\/2022 at the front right), and Maturity in years (diagonal right axis, with values visible at 20 and 0.25 years marked at the front right edge). The three-dimensional surface is rendered as a series of stacked yield curve cross-sections over time, with each thin slice representing the yield curve shape at a specific date. The color of the surface transitions from gray at the back (earliest dates in the early 1980s) through orange in the middle period (roughly 1990s\u2013early 2000s) to predominantly blue in the most recent period (2010s\u20132022), with the color variation providing additional visual differentiation across time. Early 1980s (back of the chart, gray) shows the surface begins at very high yield levels of approximately 14\u201316%, consistent with the peak interest rate environment of 1981\u20131982, with the yield curve appearing relatively flat or inverted at these extreme levels. 1980s\u20131990s (orange section) shows yields decline substantially from their early 1980s peaks, with the surface descending steeply. The yield curve takes on a more normal upward-sloping shape as short-term rates fall faster than long-term rates. 2000s\u20132010s (transition to blue) shows yields continue declining toward historically low levels, with the surface flattening considerably in the vertical dimension as rates approach zero. 2010s\u20132021 (blue, near-zero era) shows the surface compresses toward near-zero yield levels across most maturities, reflecting the extended zero interest rate policy period. The yield curve shape during this period shows a very flat profile at low absolute levels. 2022 (front of chart, orange reappearance) shows a sharp upward spike is visible at the most recent dates, reflecting the rapid Federal Reserve rate hiking cycle of 2022, with short-term yields rising dramatically and the curve potentially inverting again.<\/span><\/p>\n

Figure 7.7<\/a>: Three-dimensional surface chart displaying the evolution of the U.S. Treasury yield curve over time, rendered with a smooth continuous surface and a vibrant color gradient. The three axes are: Yield (%) on the vertical axis ranging from 0 to 16, Date on the left horizontal axis running from approximately 1985 to 2022, and Maturity on the right horizontal axis showing values of 0.5, 2, 5, 10, and 30 years. The surface is colored using a heat-map gradient transitioning from bright yellow at the highest yield levels at the back-upper portion of the surface (early 1980s, long maturities), through orange and red at intermediate yield levels, to deep purple and violet at the lowest yield levels toward the front of the chart (recent years, near-zero rates). This color gradient provides an intuitive visual indication of yield levels independent of the vertical axis. A tooltip annotation is visible on the surface reading: x: 30, y: Jul 1984, z: 13.21, indicating that the 30-year Treasury yield in July 1984 was 13.21%, one of the data points on the high-yield plateau visible at the back of the chart. 1984\u2013early 1990s (yellow-orange region) shows the surface begins at high yield levels of approximately 13\u201316% for long maturities in the mid-1980s, with the entire surface elevated. The yield curve during this period shows a relatively flat or moderately normal shape across maturities. 1990s\u20132000s (orange-red-pink region) shows the surface descends steadily as yields decline across all maturities, with the characteristic ripple pattern visible across the maturity dimension indicating recurring periods of yield curve steepening and flattening as the business cycle progresses. 2008\u20132021 (deep purple region) shows the surface compresses dramatically toward near-zero yield levels, particularly at short maturities, while long maturity yields remain slightly higher, creating visible ridges in the surface representing the persistent but low positive slope of the yield curve during the zero interest rate policy era. 2022 (front right, slight upward spike) shows a modest rise in yields is visible at the most recent dates, consistent with the beginning of the Federal Reserve rate hiking cycle.<\/span><\/p>\n

Figure 7.8<\/a>: Line chart titled “10-Year Treasury Constant Maturity Minus 2-Year Treasury Constant Maturity.” The vertical axis shows percent ranging from -3.0 to 4.0, and the horizontal axis shows dates at approximately seven-year intervals from June 1976 to September 2023. Vertical gray shaded bands mark periods of U.S. economic recessions. The chart displays the Treasury yield curve spread, the difference between the 10-year and 2-year Treasury yields, which is one of the most widely watched recession indicators. When the spread is positive, the yield curve is normal (upward sloping); when negative, the yield curve is inverted, historically a reliable predictor of recession. 1976\u20131980 shows the spread begins at approximately +1.5%, dips briefly into negative territory around 1978\u20131979 (yield curve inversion), then recovers, preceding the recession of the early 1980s. 1980\u20131982 shows a dramatic inversion to approximately -2.2% to -2.5% occurs, the deepest inversion in the entire series, coinciding with the Fed’s aggressive tightening under Volcker and preceding the severe 1981\u20131982 recession. 1983\u20131989 shows the spread recovers sharply to approximately +2.5% as the yield curve normalizes, then gradually flattens back toward zero by the late 1980s, with a brief inversion preceding the 1990\u20131991 recession. 1990\u20132000 shows the spread cycles from near zero back up to approximately +2.5% in the early 1990s recovery, then declines again toward zero and briefly negative around 1998\u20132000, preceding the 2001 recession. 2001\u20132007 shows the spread recovers to approximately +2.5%, then declines steadily toward zero and briefly inverts around 2006\u20132007, preceding the Great Recession of 2008\u20132009. 2008\u20132015 shows the spread widens dramatically to nearly +2.8% as short-term rates are cut to zero while long-term rates remain higher, then gradually narrows through 2015\u20132018. 2018\u20132019 shows the spread approaches zero and briefly inverts slightly, signaling potential recession risk before the COVID-19 pandemic. 2022\u20132023 shows the spread inverts sharply to approximately -1.0%, one of the more significant inversions in the series, reflecting the Federal Reserve’s rapid rate hiking cycle pushing short-term rates above long-term rates, consistent with elevated recession risk warnings.<\/span><\/p>\n

Figure 8.2<\/a>: Line chart titled “Real Gross Domestic Product” showing the quarterly compound annual percent change in U.S. real GDP from April 1947 to approximately 2022. The vertical axis shows compound annual percent change ranging from -40.0 to 40.0, and the horizontal axis shows dates at approximately decade intervals. Vertical gray shaded bands mark periods of U.S. economic recessions throughout the chart. The chart displays quarterly real GDP growth rates, which fluctuate considerably around a generally positive mean throughout the entire postwar period. 1947\u20131960s (high volatility era) show growth rates are highly volatile in the early postwar period, with frequent swings between approximately -10% and +17%. The amplitude of quarterly fluctuations is notably larger than in later decades. 1970s\u20131980s (stagflation and recession era) show growth remains volatile with several notable negative quarters coinciding with the oil shocks and recessions of 1973\u20131975 and 1981\u20131982, with troughs reaching approximately -10% and -8% respectively. A notable spike to approximately +16% occurs around 1978. 1990s\u20132000s (Great Moderation) shows quarterly GDP growth becomes notably less volatile, fluctuating within a narrower band of approximately -3% to +8%, reflecting the era of reduced macroeconomic volatility known as the Great Moderation. The 2008\u20132009 Great Recession produces a trough of approximately -8% to -9%. 2020\u20132021 (COVID-19 shock) shows that by far the most dramatic feature of the entire chart occurs at the far right, where the COVID-19 pandemic produces an unprecedented collapse in Q2 2020 to approximately -29% to -31% annualized, the largest quarterly decline in the entire postwar series by a wide margin, followed immediately by an equally unprecedented rebound to approximately +33% to +35% in Q3 2020 as the economy partially reopened. Growth then normalizes to more typical levels through 2021\u20132022.<\/span><\/p>\n

Figure 8.3<\/a>: Line chart displaying the quarterly compound annual percent change for three components of U.S. GDP from April 1947 to approximately 2022. The vertical axis shows percent change ranging from -100.0 to 150.0, and the horizontal axis shows dates at approximately decade intervals. A legend identifies three series. PCECC96 (blue line) represents real Personal Consumption Expenditures (PCE), the largest component of GDP. The blue line tracks closely along the zero line throughout the entire period with relatively modest fluctuations, generally ranging between approximately -5% and +10%, reflecting the relative stability of consumer spending compared to other GDP components. The line is nearly indistinguishable from the baseline for much of the chart due to its small amplitude relative to the other two series. GCEC1 (orange line) represents real Government Consumption Expenditures and Gross Investment. The orange line shows moderate volatility, generally fluctuating between approximately -10% and +65%, with the largest spikes occurring in the late 1940s and early 1950s and more modest fluctuations thereafter. Government spending volatility diminishes considerably from the 1960s onward. GPDIC1 (green line) represents real Gross Private Domestic Investment. By far the most volatile of the three series, the green line dominates the visual scale of the chart with dramatic swings throughout the entire period. Early postwar spikes reach approximately +100\u2013135% and drops to approximately -100% in the late 1940s. Throughout the series, private investment exhibits much larger cyclical swings than consumption or government spending, consistent with the well-established economic principle that investment is the most volatile component of GDP, amplifying business cycle fluctuations. The COVID-19 period produces another large spike and trough visible at the far right.<\/span><\/p>\n

Figure 8.4<\/a>:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Country<\/th>\nFRED code<\/th>\nCountry<\/th>\nFRED code<\/th>\nCountry<\/th>\nFRED code<\/th>\nCountry<\/th>\nFRED code<\/th>\n<\/tr>\n<\/thead>\n
Australia<\/td>\nAU<\/td>\nFrance<\/strong><\/td>\nFR<\/strong><\/td>\nJapan<\/strong><\/td>\nJP<\/strong><\/td>\nSlovak Republic<\/td>\nSK<\/td>\n<\/tr>\n
Austria<\/td>\nAT<\/td>\nGermany<\/strong><\/td>\nDE<\/strong><\/td>\nKorea<\/td>\nKR<\/td>\nSlovenia<\/td>\nSI<\/td>\n<\/tr>\n
Belgium<\/td>\nBE<\/td>\nGreece<\/td>\nGR<\/td>\nLuxembourg<\/td>\nLU<\/td>\nSouth Africa<\/td>\nZA<\/td>\n<\/tr>\n
Brazil<\/td>\nBR<\/td>\nHungary<\/td>\nHU<\/td>\nMexico<\/td>\nMX<\/td>\nSpain<\/td>\nES<\/td>\n<\/tr>\n
Canada<\/strong><\/td>\nCA<\/strong><\/td>\nIceland<\/td>\nIS<\/td>\nNetherlands<\/td>\nNL<\/td>\nSweden<\/td>\nSE<\/td>\n<\/tr>\n
Chile<\/td>\nCL<\/td>\nIdia<\/td>\nIN<\/td>\nNewZealand<\/td>\nNZ<\/td>\nSwitzerland<\/td>\nCH<\/td>\n<\/tr>\n
Czech Republic<\/td>\nCZ<\/td>\nIndonesia<\/td>\nID<\/td>\nNorway<\/td>\nNO<\/td>\nTurkey<\/td>\nTR<\/td>\n<\/tr>\n
Denmark<\/td>\nDK<\/td>\nIreland<\/td>\nIE<\/td>\nPoland<\/td>\nPL<\/td>\nUnited Kingdom<\/strong><\/td>\nGB<\/strong><\/td>\n<\/tr>\n
Estonia<\/td>\nEE<\/td>\nIsrael<\/td>\nIL<\/td>\nPortugal<\/td>\nPT<\/td>\nUnited States<\/strong><\/td>\nUS<\/strong><\/td>\n<\/tr>\n
Finland<\/td>\nFI<\/td>\nItaly<\/strong><\/td>\nIT<\/strong><\/td>\nRussia<\/td>\nRU<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Figure 8.5<\/a>: Screenshot of the bottom rows of a Microsoft Excel spreadsheet, showing rows 313 through 315 and columns A through H, displaying the final data row, standard deviation formulas, and calculated results for four data series. Row 313 (final data row) contains the last data entries for four paired date-value columns. Column A: 07\/01\/2023, Column B: 4.9, Column C: 07\/01\/2023, Column D: 4.0, Column E: 07\/01\/2023, Column F: 8.4, Column G: 07\/01\/2023, Column H: 4.6. All four series share the same end date of July 1, 2023, suggesting the data runs from some earlier start date through mid-2023, with row 313 being the last observation (implying approximately 308 data points per series starting from row 6). Row 314 (formula row) contains Excel STDEV.P (population standard deviation) formulas for each value column: Column B: =STDEV.P(B6:B313), Column D: =STDEV.P(D6:D313), Column F: =STDEV.P(F6:F313), Column H: =STDEV.P(H6:H313). These formulas calculate the population standard deviation across the full range of each data series. Row 315 (results row) displays the calculated standard deviation values with a “SD –>” label in Column A: Column B: 4.6, Column D: 4.4, Column F: 20.9, Column H: 7.4.<\/span><\/p>\n

Figure 10.1<\/a>: Population pyramid chart labeled “Year = 24” displaying the age-sex distribution of a population at year 24 of a simulation or projection. The horizontal axis shows population counts ranging from 30,000 on the left to 30,000 on the right, with zero at the center. Blue bars extend to the left representing one sex (typically male) and red bars extend to the right representing the other sex (typically female). The chart shows two distinct population clusters separated by a gap. The upper cluster (larger, rectangular shape) is a broad, roughly rectangular block of blue (left) and red (right) bars centered at approximately 10,000\u201312,000 individuals per bar, with relatively uniform width across several age groups. The rectangular shape suggests a cohort of similar size moving through the age structure, consistent with a baby boom or large generational cohort. The lower cluster (smaller, irregular hourglass or star shape) is a smaller, more irregular cluster below the upper group, with a distinctive pinched or hourglass shape, wider at the top and bottom of this cluster and narrower in the middle. This unusual shape suggests a cohort with uneven age distribution, possibly reflecting a baby bust, mortality event, or other demographic disruption between the two clusters.<\/span><\/p>\n

Figure 10.2<\/a>: Population pyramid chart labeled “Year = 100” displaying the age-sex distribution of a population at year 100 of a simulation or projection, serving as a companion to the Year = 24 chart. The horizontal axis shows population counts ranging from 30,000 on the left to 30,000 on the right, with zero at the center. Blue bars extend to the left representing one sex (typically male) and red bars extend to the right representing the other sex (typically female). By year 100 the population pyramid has evolved dramatically from the fragmented, two-cluster structure visible at year 24 into a single continuous, broadly triangular or roughly Christmas-tree-shaped distribution spanning the full vertical height of the chart. There are a few key features of the year 100 pyramid. The base (youngest age groups) are where the widest bars appear near the bottom, extending to approximately 25,000\u201328,000 on each side, indicating a large young population. Middle age groups show the pyramid narrows gradually and relatively smoothly moving upward through middle age cohorts, with modest irregularities in the mid-section, including a slight indentation or notch visible on both sides at approximately one-third of the way up. The upper age groups (oldest) shows the pyramid tapers to near zero at the top, reflecting normal mortality attrition among the oldest age groups. The overall shape shows the broadly triangular shape with a wide base and narrowing apex is characteristic of a growing population with relatively high birth rates and normal age-specific mortality. The two-sided symmetry between blue and red is approximate, consistent with roughly equal sex ratios at birth and similar survival patterns.<\/span><\/p>\n

Figure 10.5<\/a>: Line chart titled “United States Dependency Ratio” showing the number of dependents per working-age adult from 1980 to 2060, with historical data and projections. The vertical axis shows dependents per working-age adult ranging from 0 to 0.70, and the horizontal axis shows years from 1970 to 2070. Two line segments are plotted in different colors to distinguish historical data from projections. The blue line (historical data, approximately 1980\u20132024) shows the dependency ratio begins at approximately 0.51 in 1980, remains relatively flat through the mid-1980s, rises modestly to approximately 0.53 around 1990, then declines gradually through the 1990s and 2000s to a trough of approximately 0.49 around 2010, reflecting the working-age Baby Boomer generation at peak employment. The ratio then rises from approximately 2010 onward, accelerating through the 2010s and early 2020s as Baby Boomers begin retiring, reaching approximately 0.57\u20130.58 by approximately 2023\u20132024. The orange line (projected data, approximately 2024\u20132060) shows the projection continues the upward trend steeply from the historical endpoint, rising sharply to approximately 0.62\u20130.63 by 2030, then leveling off and remaining relatively flat between approximately 0.62 and 0.65 through the 2030s and 2040s, before rising slightly again to approximately 0.66 by 2060.<\/span><\/p>\n

Figure 10.6<\/a>: Line chart titled “United States Old-age Dependency Ratio” showing the number of elderly dependents per working-age adult from 1980 to 2060, with historical data and projections. The vertical axis shows dependents per working-age adult ranging from 0 to 0.45, and the horizontal axis shows years from 1970 to 2070. Two line segments are plotted in different colors to distinguish historical data from projections. The blue line (historical data, approximately 1980\u20132024) shows the old-age dependency ratio begins at approximately 0.17 in 1980 and remains relatively flat through the 1980s at approximately 0.17\u20130.19, reflecting the relatively small pre-Baby Boomer elderly population during this period. The ratio stays nearly flat through the 1990s at approximately 0.19\u20130.20, then begins a gradual rise through the 2000s as early retirees enter the elderly population. From approximately 2010 onward the rate of increase accelerates markedly, rising from approximately 0.19 in 2010 to approximately 0.25 by 2020 and approximately 0.29\u20130.30 by approximately 2024, reflecting the accelerating retirement of the large Baby Boomer generation. The orange line (projected data, approximately 2024\u20132060) shows the projection continues the steep upward trajectory, rising sharply to approximately 0.35 by 2030, then leveling off and plateauing at approximately 0.35\u20130.36 through the mid-2030s to mid-2040s as the Baby Boomer retirement wave completes. The ratio then resumes a gradual upward trend, reaching approximately 0.39\u20130.40 by 2060, reflecting ongoing increases in longevity and the aging of subsequent generations.<\/span><\/p>\n","protected":false},"author":65,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"back-matter-type":[],"contributor":[],"license":[],"_links":{"self":[{"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/pressbooks\/v2\/back-matter\/604"}],"collection":[{"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/wp\/v2\/users\/65"}],"version-history":[{"count":8,"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/pressbooks\/v2\/back-matter\/604\/revisions"}],"predecessor-version":[{"id":623,"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/pressbooks\/v2\/back-matter\/604\/revisions\/623"}],"metadata":[{"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/pressbooks\/v2\/back-matter\/604\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/wp\/v2\/media?parent=604"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/pressbooks\/v2\/back-matter-type?post=604"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/wp\/v2\/contributor?post=604"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.palni.org\/gatewaytobusinessanalytics\/wp-json\/wp\/v2\/license?post=604"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}