Problem 12 - Applied Calculus

Question

The consumer price index (CPI) for a given year is the amount of money in that year that has the same purchasing power as \(\$ 100\) in 1983 . At the start of 2009 , the CPI was \(211 .\) Write a formula for the CPI as a function of \(t\) years after \(2009,\) assuming that the CPI increases by \(2.8 \%\) every year.

Step-by-Step Solution

Answer

The CPI formula is \( C(t) = 211 \times 1.028^t \).

Step 1: Understand the Problem

The Consumer Price Index (CPI) in 2009 is given as 211, which indicates the value equivalent to $100 in 1983. We need to find a formula representing CPI as a function of years, assuming a 2.8% yearly increase.

Step 2: Define the Variables

Let \( C(t) \) represent the CPI as a function of \( t \), where \( t \) is the number of years after 2009. The CPI in 2009, which is \( C(0) \), is 211.

Step 3: Determine Growth Formula

Since the CPI increases at an annual rate of 2.8%, we use the formula for exponential growth. The general formula is: \( C(t) = C_0 \times (1 + r)^t \) where \( C_0 \) is the initial CPI in 2009, and \( r \) is the growth rate expressed as a decimal.

Step 4: Apply the Growth Formula

Substitute the initial conditions and growth rate into the formula: \( C(t) = 211 \times (1 + 0.028)^t \).

Step 5: Simplify the Formula

Simplify the formula to: \( C(t) = 211 \times 1.028^t \). This formula allows you to find the CPI for any year after 2009.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases by a constant factor over equal intervals of time. This concept is clearly demonstrated in the Consumer Price Index (CPI) problem, where the CPI increases annually by a fixed percentage. In exponential growth, each year's growth builds upon the previous year's, leading to compound increases over time.
In the formula used to calculate CPI, exponential growth is expressed as:

• \(C(t) = C_0 \times (1 + r)^t\)

This formula indicates that the CPI in year \(t\) is calculated by multiplying the initial CPI by the growth factor \((1 + r)^t\), where \(r\) is the growth rate as a decimal.
The compound nature of exponential growth means that as the time period \(t\) increases, the value of \((1 + r)^t\) grows increasingly larger, showcasing the power of compound increases.

Function of Time

The concept of a function of time is used to model how a particular value changes over a given period. In the context of the CPI, it forms a mathematical representation that connects the CPI to the passage of years since a certain starting point, 2009 in this case.
The function expression for CPI over time is:

• \(C(t) = 211 \times 1.028^t\)

Here, \(t\) represents years after 2009.
This formula helps us understand how the CPI evolves, providing a predictive model for future CPI based on the predefined growth rate.

• \(C(0) = 211\), representing 2009.
• \(t\) aligns directly with the number of years passing (e.g., \(t = 1\) for 2010).

This functional relationship allows economists and analysts to predict economic trends and make informed decisions based on expected future values.

Percentage Increase

Understanding percentage increase is crucial for interpreting changes like those in the CPI. A percentage increase refers to how much a value grows relative to its original size over a period.
To compute the new value after a percentage increase:

• Multiply the original amount by one plus the percentage increase expressed as a decimal.

For the CPI, an annual growth rate of 2.8% equates to:

• A growth factor of \((1 + 0.028)\), since 2.8% as a decimal is 0.028.

This compound increase means each year, the new CPI is 2.8% higher than the last.

• For an initial CPI of 211, each year it increases multiplicatively by a factor of 1.028.
• This helps to model the compounded effect of the annual percentage increase over time accurately.

By grasping the percentage increase, one can apply this understanding to various financial and economic scenarios beyond just the CPI.