Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Devaluation: Due to inflation, a dollar loses its purchasing power with time. Suppose that the dollar loses its purchasing power at a rate of \(2 \%\) per year. (a) Find a formula that gives us the purchasing power of \(\$ 1 t\) years from now. (b) Use your calculator to approximate the number of years it will take for the purchasing power of the dollar to be cut in half.

Step-by-Step Solution

Answer

The purchasing power of one dollar \(t\) years from now is given by the formula \(P(t) = 0.98^t\). It will take approximately 35 years for the purchasing power of the dollar to be cut in half.

Step 1: Determining Formula to calculate purchasing power

Start by formulating a mathematical representation for devaluation. If the dollar loses \(2\%\) of its purchasing power each year, this can be represented as \(1 - 0.02 = 0.98\). This suggests that each year, a dollar retains \(98\%\) of its previous year's worth. Thus, after \(t\) years, a dollar will still be worth \(0.98^t\). So, the formula that gives the purchasing power of $1 t years from now is \(P(t) = 0.98^t\).

Step 2: Calculate the number of years with this formula

To find the number of years it will take for the purchasing power of the dollar to be cut in half, we set \(0.98^t = 0.5\) and solve for \(t\). This is an exponential equation, so to solve it, we will take the logarithm of both sides.

Step 3: Using logarithms to solve the equation

Taking the natural logarithm of both sides, we get \( \ln{0.98^t} = \ln{0.5}\). Applying the logarithm rule this can be simplified as \(t \cdot \ln{0.98} = \ln{0.5}\). Consequently, \(t = \frac{\ln{0.5}}{\ln{0.98}}\). Using the calculator, evaluate this expression to find the desired number of years.

Key Concepts

Purchasing Power

Purchasing power describes the ability of a currency to buy goods and services. This concept is crucial because it determines how much your money is truly worth.
If the purchasing power of money decreases, it means you cannot buy as much with the same amount of money as before. Over time, due to various factors like inflation, the purchasing power of money often decreases.

• When we talk about purchasing power, it's essential to think of it in real terms, not just the nominal amount you see in your bank account.
• The relationship between nominal value and real value is critical for understanding economic conditions.

Understanding purchasing power helps you make smarter financial decisions, especially in planning investments, savings, and overall financial well-being.

Inflation

Inflation is a measure of the rate at which the general level of prices for goods and services is rising, leading to a decrease in purchasing power.
Inflation impacts everyone's daily life, as the cost of living increases when prices rise. Here are some key points to understand about inflation:

• Inflation is usually expressed as a percentage, such as a 2% increase per year.
• It can be influenced by various factors including demand for goods and services, production costs, and government policies.

Being aware of inflation is crucial because it affects how much you need to spend to maintain your standard of living.
Planning for inflation involves investing wisely and saving money, to help preserve your purchasing power over time.

Exponential Functions

Exponential functions are used to describe many natural processes, including the growth or decay of quantities. In finance, exponential decay helps us understand how things like purchasing power can decrease over time.

• An exponential function can be written in the form \(a^t\), where \(a\) is a constant and \(t\) is the time period.
• For your exercise, the function \(P(t) = 0.98^t\) represents how much a dollar retains of its purchasing power over time.

Here, 0.98 is the base which shows 98% of the value is retained each year, indicating a 2% loss. Understanding exponential functions allows you to model real-world scenarios effectively, giving you greater insight into financial trends and predictions.

Logarithms

Logarithms are the inverse operations to exponentiation and are very useful when dealing with exponential equations. They provide a way to solve for unknown exponents in equations of the form \(a^t = b\).

• In the context of the problem, we use logarithms to determine the number of years it takes for the purchasing power to decrease by a certain factor.
• By taking the natural log of both sides of \(0.98^t = 0.5\), we can solve for \(t\).

This involves using the property that says \(\ln a^b = b \ln a\). By rearranging the equation, we find that \(t = \frac{\ln{0.5}}{\ln{0.98}}\).
Logarithms help simplify the process of solving otherwise complex exponential equations, making them a powerful tool in mathematics and finance.