Problem 62
Question
In the short run, per capita personal income (PCPI) in the United States grows approximately linearly. In 2009 PCPI was 38.6, and in 2012 it had grown to 42.8 (both in thousands of dollars). a. Use the two (year, \(\mathrm{PCPI}\) ) data points (1,38.6) and (4,42.8) to find the linear relationship \(y=m x+b\) between \(x=\) years since 2008 and \(y=\mathrm{PCPI}\). b. Interpret the slope of the line. c. Use your linear relationship to predict \(\mathrm{PCPI}\) in 2020 .
Step-by-Step Solution
Verified Answer
a) y = 1.4x + 37.2; b) PCPI increases by $1,400 per year; c) In 2020, PCPI is predicted to be 54,000.
1Step 1: Identify the Given Data Points
Given two data points: \((1, 38.6)\) and \((4, 42.8)\). Here, \(x\) represents the number of years since 2008 and \(y\) is the per capita personal income (PCPI) in thousands of dollars.
2Step 2: Find the Slope (m)
The slope \(m\) of the line is calculated using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points: \(m = \frac{42.8 - 38.6}{4 - 1} = \frac{4.2}{3} = 1.4\).
3Step 3: Write the Linear Equation
The equation of the line is \(y = mx + b\). Substitute \(m = 1.4\) and use one of the points to find \(b\). Using point \((1, 38.6)\): \(38.6 = 1.4 \cdot 1 + b\). Solve for \(b\): \(b = 38.6 - 1.4 = 37.2\). Thus, the linear equation is \(y = 1.4x + 37.2\).
4Step 4: Interpret the Slope
The slope \(m = 1.4\) represents the rate of increase in PCPI per year since 2008. This means PCPI increases by $1,400 per year.
5Step 5: Predict PCPI in 2020
First, compute the value of \(x\) for the year 2020, which is \(2020 - 2008 = 12\). Substitute \(x = 12\) into the equation \(y = 1.4x + 37.2\): \(y = 1.4 \cdot 12 + 37.2 = 16.8 + 37.2 = 54\).
Key Concepts
Linear Relationships in EconomicsSlope Interpretation: Understanding Rate of ChangeIncome Prediction with Linear ModelsMathematical Modeling in Economics
Linear Relationships in Economics
In economics, the concept of linear relationships is often used to model the behavior of variables over time. A linear relationship indicates that as one variable changes, another changes in a consistent manner.
This is described using the equation of a line:
When two data points are provided, such as \((1, 38.6)\) and \((4, 42.8)\), they can be used to determine the linear relationship that predicts future values. Establishing a linear relationship helps to understand how a change in one aspect, like time, can affect another aspect, such as income in this case. Understanding this relationship is fundamental in making predictions and decisions based on data trends.
This is described using the equation of a line:
- \(y = mx + b\)
When two data points are provided, such as \((1, 38.6)\) and \((4, 42.8)\), they can be used to determine the linear relationship that predicts future values. Establishing a linear relationship helps to understand how a change in one aspect, like time, can affect another aspect, such as income in this case. Understanding this relationship is fundamental in making predictions and decisions based on data trends.
Slope Interpretation: Understanding Rate of Change
In mathematics, the slope of a line in a linear relationship provides critical insights into how the dependent variable changes with the independent variable. The slope, represented by \(m\) in the equation \(y = mx + b\), defines the rate of change.
In our exercise, the slope is calculated between the two points: \( (1, 38.6) \) and \( (4, 42.8) \). The formula for calculating slope is:
This means that for every year that passes, the per capita personal income (PCPI) increases by $1,400.
Understanding the slope is crucial in economics as it indicates the direction and rate of financial changes, helping economists to draw conclusions and make forecasts.
In our exercise, the slope is calculated between the two points: \( (1, 38.6) \) and \( (4, 42.8) \). The formula for calculating slope is:
- \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
This means that for every year that passes, the per capita personal income (PCPI) increases by $1,400.
Understanding the slope is crucial in economics as it indicates the direction and rate of financial changes, helping economists to draw conclusions and make forecasts.
Income Prediction with Linear Models
Predicting future trends is a powerful capability of linear equations. By using a linear relationship, economists can estimate future values based on past data. In this context, predicting the per capita personal income (PCPI) involves substituting future years into the established linear equation.
With the equation \(y = 1.4x + 37.2\), predicting PCPI in 2020 involves understanding that 2020 is 12 years after 2008 (hence \(x = 12\)).
Substituting \(x\) into the linear equation:
This predictive power is valuable for planning and policy-making, offering a glimpse into future economic conditions.
With the equation \(y = 1.4x + 37.2\), predicting PCPI in 2020 involves understanding that 2020 is 12 years after 2008 (hence \(x = 12\)).
Substituting \(x\) into the linear equation:
- \(y = 1.4 \cdot 12 + 37.2\)
This predictive power is valuable for planning and policy-making, offering a glimpse into future economic conditions.
Mathematical Modeling in Economics
Mathematical modeling involves creating equations to represent real-world scenarios mathematically. In economics, this enables the understanding and forecasting of commodities like income, prices, and economic growth. By choosing the right model, economists can transform complex real-world situations into understandable and actionable insights.
In our exercise, the linear model \(y = 1.4x + 37.2\) creates a simplified representation of how PCPI changes over time.
This model is preferred because it is easy to understand and calculate, offering a straightforward illustration of the trend over a given period. When plotted, the line visually demonstrates the relationship and facilitates decision-making based on its trajectory.
Mathematical models, like linear ones, are indispensable tools in economic analysis, providing structured approaches to predicting future scenarios based on existing data.
In our exercise, the linear model \(y = 1.4x + 37.2\) creates a simplified representation of how PCPI changes over time.
This model is preferred because it is easy to understand and calculate, offering a straightforward illustration of the trend over a given period. When plotted, the line visually demonstrates the relationship and facilitates decision-making based on its trajectory.
Mathematical models, like linear ones, are indispensable tools in economic analysis, providing structured approaches to predicting future scenarios based on existing data.
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