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Chapter 4: Problem 9

Inflation: The yearly inflation rate tells the percentage by which prices increase. For example, from 1990 through 2000 the inflation rate in the United States remained stable at about \(3 \%\) each year. In 1990 an individual retired on a fixed income of \(\$ 36,000\) per year. Assuming that the inflation rate remains at \(3 \%\), determine how long it will take for the retirement income to deflate to half its 1990 value. (Note: To say that retirement income has deflated to half its 1990 value means that prices have doubled.)

Short Answer

Expert verified
It will take approximately 23.45 years for the retirement income to deflate to half its 1990 value with a 3% annual inflation rate.

Step by step solution

01

Understanding Inflation and Deflation

Inflation at a rate of 3% annually means that the prices of goods and services increase by 3% each year. The retirement income deflating to half its 1990 value implies that prices have doubled.
02

Determining the Time for Prices to Double

Set up the equation based on the concept of inflation compounding yearly: if prices increase by 3% each year, the price level after \(n\) years would be \((1.03)^n\) times the initial price level. We want \((1.03)^n = 2\) because the prices need to double.
03

Solving the Exponential Equation

To solve \((1.03)^n = 2\), use logarithms:\[ n \cdot \log(1.03) = \log(2) \] Dividing both sides by \(\log(1.03)\) gives:\[ n = \frac{\log(2)}{\log(1.03)} \]
04

Calculating the Number of Years

Calculate the right side of the equation:\[ n \approx \frac{0.3010}{0.0128} \approx 23.45 \]This means it will take approximately 23.45 years for the retirement income to deflate to half its value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Growth
Understanding exponential growth is crucial for comprehending inflation and deflation over time. Exponential growth occurs when a quantity increases by a consistent percentage over equal time periods. In the context of inflation, prices grow exponentially when they increase by a fixed percentage consistently every year. For instance, a 3% annual increase in prices means every subsequent year's cost is 103% of the previous year. This compound effect is represented mathematically in the equation \((1.03)^n\), where 1.03 represents a 3% increase and \(n\) is the number of years. Exponential growth forms the basis of inflation models, where each year builds upon the last, leading to a rapid increase over time. This can significantly impact purchasing power, making it essential for financial planning.
Logarithmic Equations
Logarithmic equations are crucial tools in solving problems involving exponential growth, such as inflation. When you have an equation like \((1.03)^n = 2\), and you need to find an unknown exponent \(n\), logarithms are your go-to tool. Here, the equation \((1.03)^n = 2\) signifies that prices have doubled.Logarithms can transform the exponential equation into a linear one:\[ n \cdot \log(1.03) = \log(2) \] This simplification helps isolate the variable \(n\), allowing you to solve for it explicitly. By moving from an exponential equation to a logarithmic form, you use the properties of logarithms that convert multiplication into addition or division, making it possible to find the solution more directly. Understanding logarithms provides a powerful method to grapple with exponential phenomena effectively.
Financial Modeling
Financial modeling uses mathematical representations of real-world financial situations to predict future outcomes. In our inflation example, financial modeling helps determine the impact of a steady 3% inflation rate on purchasing power over time. By setting up equations based on initial conditions and growth rates, analysts can predict future scenarios.Through financial modeling, one can answer questions like "How long until my retirement income halves in value?" as shown by the calculation \(n \approx \frac{0.3010}{0.0128} \approx 23.45\). This example illustrates an individual's fixed retirement income against the backdrop of inflation. The model helps visualize the loss of purchasing power鈥攅ssential for retirement planning. Financial modeling is more than numbers; it reflects the economic realities underlining crucial decisions.

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Most popular questions from this chapter

An exponential model with unit adjustment: Show that the following data are exponential and find a formula for an exponential model. (Note: It will be necessary to make a unit adjustment. For this problem, round your answers to three decimal places.) $$ \begin{array}{|c|c|c|c|} \hline t & g(t) & t & g(t) \\ \hline 0 & 38.30 & 12 & 16.04 \\ \hline 4 & 28.65 & 16 & 11.99 \\ \hline 8 & 21.43 & 20 & 8.97 \\ \hline \end{array} $$

Growth rate: An animal grows according to the formula $$ L=0.6 \log (2+5 T) . $$ Here \(L\) is the length in feet and \(T\) is the age in years. a. Draw a graph of length versus age. Include ages up to 20 years. b. Explain in practical terms what \(L(15)\) means, and then calculate that value. c. How old is the animal when it is 1 foot long? d. Explain in practical terms what the concavity of the graph means. e. Use a formula to express the age as a function of the length.

Stochastic population growth: Many populations are appropriately modeled by an exponential function, at least for a limited period of time. But there are many factors contributing to the growth of any population, and many of them depend on chance. There are a number of ways to produce stochastic models. We consider one that is an illustration of a simple Monte Carlo method. We want to model a population that is initially 500 and grows at an average rate of 2% per year. To do this we make a table of values for population according to the following procedure. The first entry in the table is for time t = 0, and it records the initial value 500. To get the entry corresponding to t = 1, we roll a die and change the population according to the face that appears, using the following rule. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Face } \\ \text { appearing } \end{array} & \begin{array}{c} \text { Population } \\ \text { change } \end{array} \\ \hline 1 & \text { Down 2\% } \\ \hline 2 & \text { Down } 1 \% \\ \hline 3 & \text { No change } \\ \hline 4 & \text { Up 2\% } \\ \hline 5 & \text { Up 4\% } \\ \hline 6 & \text { Up 9\% } \\ \hline \end{array} $$ To get the entry corresponding to \(t=2\), we roll a die and change the population from \(t=1\), again using the above rule. This procedure is then followed for \(t=3\), and so on. a. Using this procedure, make a table recording the population values for years 0 through 10 . b. Plot the data points from your table and the exponential model on the same screen. Comment on the level of agreement.

State quakes: The largest recorded earthquake centered in Idaho measured \(7.2\) on the Richter scale. a. The largest recorded earthquake centered in Montana was \(3.16\) times as powerful as the Idaho earthquake. What was the Richter scale reading for the Montana earthquake? b. The largest recorded earthquake centered in Arizona measured \(5.6\) on the Richter scale. How did the power of the Idaho quake compare with that of the Arizona quake?

Rice production in Asia: In Exercise 16 of Section 3.4, it was estimated that annual rice yield in Asia was increasing by about \(0.04\) ton per hectare each year. The International Rice Research Institute \(^{1}\) predicts an increase in demand for Asian rice of about \(2.1 \%\) per year. Discuss the future of rice production in Asia if these estimates remain valid.
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