Problem 92 - Brief Applied Calculus

Question

ECONOMICS: Disposable Income Per capita disposable income (that is, after taxes have been subtracted) has fluctuated significantly in recent years, as shown in the following table. \begin{tabular}{lccccc} \hline Year & 2005 & 2006 & 2007 & 2008 & 2009 \\ \hline Percentage Change & \(0 \%\) & \(3.7 \%\) & \(5.6 \%\) & \(5.3 \%\) & \(2.7 \%\) \\\ \hline \end{tabular} a. Number the data columns with \(x\) -values \(1-5\) (so that \(x\) stands for years since 2004 ) and use quadratic regression to fit a parabola to the data. State the regression function. [Hint: See Example 10.] b. Use your regression curve to estimate the year and month when the percentage change was at its maximum.

Step-by-Step Solution

Answer

a. The regression function is \( y = -0.54x^2 + 3.50x - 2 \). b. The maximum change occurred around April 2007.

Step 1: Assign x-values

We start by assigning x-values to each year. For this, since the table represents years from 2005 to 2009, we set: 2005 as \( x = 1 \), 2006 as \( x = 2 \), 2007 as \( x = 3 \), 2008 as \( x = 4 \), and 2009 as \( x = 5 \).

Step 2: Prepare Data for Regression

Create pairs of \((x, y)\) values where \(x\) is the year number (1 through 5) and \(y\) is the percentage change: \((1, 0)\), \((2, 3.7)\), \((3, 5.6)\), \((4, 5.3)\), and \((5, 2.7)\).

Step 3: Perform Quadratic Regression

Using a statistical calculator or software, input the pairs \((x, y)\) to perform quadratic regression. This will yield a quadratic equation of the form: \( y = ax^2 + bx + c \).

Step 4: Determine Regression Function

After performing quadratic regression with the data, you should derive the quadratic function: \( y = -0.54x^2 + 3.50x - 2 \) (note: the exact coefficients may slightly vary depending on rounding).

Step 5: Find the Maximum of the Quadratic Function

The parabola's vertex will give the maximum percentage change because the quadratic term \(a\) is negative. Use the vertex formula \( x = -\frac{b}{2a} \). Substitute \(a = -0.54\) and \(b = 3.50\) to find: \( x = \frac{-3.50}{-1.08} \approx 3.24 \).

Step 6: Estimate the Date of Maximum Change

Round \(x = 3.24\) to the nearest year (since we are dealing with annual data): approximately the 3rd year. Since our x-values start in 2005, \(x = 3.24\) indicates the year 2007. The .24 portion of the year suggests slightly after March, roughly April 2007.

Key Concepts

Disposable Income

Disposable income refers to the amount of money an individual or household has available to spend and save after accounting for taxes. It serves as an indicator of financial health and a critical factor in economic analyses. In simple terms, it's what you actually get to take home.

• It affects consumer spending, which is a significant driver of economic growth.
• Economists often analyze changes in disposable income to predict spending trends.
• Factors influencing disposable income include tax rates, inflation, and wage changes.

Understanding disposable income helps individuals plan their expenses and savings more effectively. It's like knowing how much is left in your wallet after the necessary deductions have been taken out.

Percentage Change

Percentage change is a way to express a change in value as a percentage of the original value. It's crucial in understanding economic data because it provides context to changes: how significant or insignificant they really are.

• This is calculated by the formula: \( \text{Percentage Change} = \left( \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \right) \times 100 \% \).
• A positive percentage indicates an increase, while a negative percentage indicates a decrease.
• This measure is particularly useful when comparing economic indicators over different periods.

In the exercise, it helped quantify how disposable income changed each year, giving a clear picture of economic trends.

Vertex Formula

The vertex formula is a crucial component in analyzing quadratic functions. It allows you to find the highest or lowest point on the parabola, known as the vertex. In the context of quadratic regression, it helps determine when a maximum or minimum value occurs.

• For a quadratic equation \( y = ax^2 + bx + c \), the x-coordinate of the vertex is \( x = -\frac{b}{2a} \).
• This formula is especially useful in economic models to find optimal points, like maximum profit or maximum change.
• In this exercise, we used it to predict the year when the percentage change in disposable income was at its peak.

It's like finding the peak of a mountain in a range; knowing the vertex helps locate the exact peak of changes over time.