Question

Inflation If the rate of inflation is \(4 \frac{1}{2} \%\) per year and the average price of a car is currently \(\$ 16,000,\) the average price after \(n\) years is \(P_{n}=\$ 16,000(1.045)^{n}\) Compute the average prices for the next 5 years.

Short answer

The average car prices for the next five years will be approximately \$16,720, \$17,472, \$18,258, \$19,090, and \$19,969 respectively.

Step-by-step solution

Understanding Inflation Rate

The inflation rate is \(4 \frac{1}{2} \%\) or \(0.045\) in decimal form. This is added to \(1\) to calculate the total yearly increase, obtaining \(1.045\).

Calculate Price After 1 Year

The formula \(P_n = \$16,000(1.045)^n\) is used. Number \(n\) represents the year, and in this case, we want to find the price after the first year, so \(n = 1\). \(P_1 = \$16,000(1.045)^1 = \$16,720\)

Calculate Price After 2 Years

For the second year, \(n = 2\). So, \(P_2 = \$16,000(1.045)^2 = \$17,472\)

Calculate Price After 3 Years

For the third year, \(n = 3\). So, \(P_3 = \$16,000(1.045)^3 = \$18,258.24\)

Calculate Price After 4 Years

For the fourth year, \(n = 4\). So, \(P_4= \$16,000(1.045)^4 = \$19,089.61\)

Calculate Price After 5 Years

For the fifth year, \(n = 5\). So, \(P_5 = \$16,000(1.045)^5 = \$19,968.65\)

Key concepts

Inflation Adjustment Formula

Understanding the inflation adjustment formula is crucial for anyone looking to comprehend how the value of money changes over time. Inflation refers to the general increase in prices and fall in the purchasing power of money. To adjust the price of an item, such as a car, and predict future costs, we use a simple yet effective formula:

\( P_{n} = P_0(1 + r)^n \)

where \( P_{n} \) is the future price after \( n \) years, \( P_0 \) is the current price, \( r \) is the inflation rate in decimal form, and \( n \) is the number of years into the future you want to predict. In our car example, the current price of \( P_0 \) is \$16,000, with an inflation rate \( r \) of 4.5% or 0.045, when calculating future prices.

Exponential Growth in Economics

When we talk about exponential growth in economics, we typically refer to a situation where the growth rate of a value, such as the price of an asset or even an entire economy, increases at a constant percentage rate over time. This differs from linear growth, where the increase is a constant amount each period.

In the context of inflation, when prices rise exponentially, they follow a pattern where each year's increase builds on the previous year's end value. This type of growth can be expressed mathematically as:

\( P_{n} = P_0e^{rt} \)

However, in most practical everyday cases, like our textbook problem, it's simplified to a formula similar to the compounding interest formula.

Time Value of Money

The time value of money is foundational to understanding the inflation adjustment formula. It tells us that a dollar today is worth more than a dollar tomorrow. This is because money available at the present time is worth more due to its potential earning capacity, which is encapsulated in the concept of interest.

This principle underlies the reason we calculate future prices with inflation. By adjusting for inflation, we are recognizing that money will not have the same purchasing power in the future. The formulas we use for calculating future value in economics are built on this very concept.

Compounding Interest Formula

The compounding interest formula closely relates to the inflation adjustment formula. In essence, compounding interest allows you to earn interest on both the money you've originally invested and the interest you've accumulated over time. The same concept applies when prices inflate over multiple periods.

The formula to calculate the compound amount is:

\( A = P(1 + r/n)^{nt} \)

Where \( A \) represents the future value of the investment/loan, including interest, \( P \) is the principal amount, \( r \) is the annual interest rate (decimal), \( n \) is the number of times that interest is compounded per year, and \( t \) is the time the money is invested or borrowed for. When inflation rates are compounded yearly, the formula simplifies, as seen in the inflation rate calculation, making \( n \) equal to 1.