Exponential & Log Models

  1. Total personal income (TPI), is a measure of the sum-total income received by individuals from all sources. The BEA reports historical data for total personal income (in trillions of dollars) in the US according to this table:
    Years since 1970 10 15 20 25 30 35 40 45 50
    TPI (in trillions) 2.32 3.52 4.91 6.28 8.62 10.55 12.56 15.47 19.63
    This TPI data can be modelled by the exponential function \(P(t) = 1.76(1 + 0.05)^t,\) where \(t\) is the number of years since 1970.
    1. What is the value of \(P(45)?\) What does this value represent within the context of the model? By what percent does this value differ from the figure reported by the BEA?
    2. Within the context of the model, what do the parameters \(1.76\) and \(0.05\) represent?
    3. What does the model predict total personal income in the US to be next year?
    4. The data is reprinted here for convenience. The equation is \(P(t) = 1.76(1 + 0.05)^t.\)
      Years since 1970 10 15 20 25 30 35 40 45 50
      TPI (in trillions) 2.32 3.52 4.91 6.28 8.62 10.55 12.56 15.47 19.63
      Sketch a scatter-plot of the data along with the graph \(y = P(t)\) on these axes to visually verify that \(P\) is a reasonable model for the data. Protip: plan your axis scaling first.
    5. Exactly what year does the model predict the US will have a total personal income of $25 trillion? What about $100 trillion? What about $1 quadrillion?
  2. The table below gives the projected life expectancy (life span) of a person born in the United States for select birth years from 1920 and through 2018 according to the CDC.
    Years since 1900 20 30 40 50 60 70 80 90 100 104 110 115 118
    Life Span 54.1 59.7 62.9 68.2 69.7 70.8 73.7 75.4 76.8 77.5 78.7 78.7 78.8
    1. Using technology, perform regression to find a logarithmic function \(f(t) = a + b\ln(t)\) that models the data, with \(t\) being the number of years since 1900.
    2. Using technology, perform regression to find a power model \(g(t) = a\cdot t^b\) that models the data, with \(t\) being the number of years since 1900.

    3. Using technology, plot the graphs of each of these functions along with the data on the same set of axis and choose which model you think fits the data best. On your plot make sure the domain of \(t\) matches up with the years 1920–2030.

      Between these two models, which one predicts life expectancy will increase more in the future? I.e. which of these models is more optimistic?

    4. What does the model you chose predict your own life expectancy to be?
    5. What does the model you chose predict the life expectancy of a baby born today to be?
    6. What year does the logarithmic model predict life expectancy to be 80 years old?

    7. In the years since 2018, life expectancy in the US has started to decline. Someone born in the US in 2019, 2020, and 2021 has a life expectancy of 79 years, 77 years, and 76.1 years respectively . Add this new data to the data set.

      Both logarithmic and power functions are increasing functions. Since life expectancy is decreasing in recent years though, these functions may no longer provide the most accurate models. What’s a type of function that more accurately models data that initially increases, but then begins to decrease? Using technology, perform regression to find such function \(h(t)\) that models the data, again with \(t\) equal to \(0\) in 1900.

    8. What does this new model predict the life expectancy of a baby born today to be? How does this compare to the figure predicted by either your logarithmic or power model?
    9. What sort of function do you think should be used to model this data with the aim of accurately predicting future life expectancy? Another option you may want to consider is a logistic (different than logarithmic) model; look up a picture of the graph of a logistic function.

    10. Research

      Notably, data for the last few years are absent from the table. Can you find reliable life expectancy data for the last few years?