Consumer Price Index The Consumer Price Index (CPI) is a...

Consumer Price Index The Consumer Price Index (CPI) is a measure of the change in the cost of goods over time. The index was 100 for the three-year period centered on 1983. For simplicity, we will assume that the CPI was exactly 100 in 1983. Then the CPI of 215.3 in 2008 indicates that an item that cost $1.00 in 1983 would cost $2.15 in 2008. The CPI has been increasing approximately linearly over the last few decades. Source: Time Almanac 2010. a. Use this information to determine an equation for the CPI in terms of , which represents the years since 1980. b. Based on the answer to part a, what was the predicted value of the CPI in 2000? Compare this estimate with the actual CPI of 172.2. c. Describe the rate at which the annual CPI is changing.

Consumer Price Index The Consumer Price Index (CPI) is a measure of the change in the cost of goods over time. The index was 100 for the three-year period centered on 1983. For simplicity, we will assume that the CPI was exactly 100 in 1983. Then the CPI of 215.3 in 2008 indicates that an item that cost $1.00 in 1983 would cost $2.15 in 2008. The CPI has been increasing approximately linearly over the last few decades. Source: Time Almanac 2010.
a. Use this information to determine an equation for the CPI in terms of , which represents the years since 1980.
b. Based on the answer to part a, what was the predicted value of the CPI in 2000? Compare this estimate with the actual CPI of 172.2.
c. Describe the rate at which the annual CPI is changing.

Key Concepts

Rate of Change

The rate of change is a fundamental concept in any linear model, quantifying how much a variable increases or decreases per unit of time. In the context of inflation measured by the CPI, it reflects the annual increase (or decrease) in price levels, offering insights into economic trends and potential adjustments in policy or personal finances.

Intercept

The intercept in a linear equation indicates the starting value of the dependent variable when the independent variable is zero. In economic models, the intercept gives a baseline value, such as the CPI at an initial reference year, from which subsequent changes are measured.

Interpolation and Extrapolation

Interpolation and extrapolation are techniques used with linear models to estimate values at points within and outside the range of known data, respectively. Interpolation involves making predictions between two known data points, while extrapolation extends the trend beyond the available data, both of which are crucial for forecasting in economic analysis.

Linear Function in Economic Modeling

A linear function is a mathematical model that describes a constant rate of change between two variables—in this case, time and the CPI. In economic contexts, linear models are often used to simplify and approximate trends, enabling predictions about future behavior based on past data.

Consumer Price Index (CPI)

The Consumer Price Index is an economic indicator that measures the average change over time in the prices paid by consumers for a basket of goods and services. It is used to assess inflation and the purchasing power of money, serving as a crucial tool for economic policy and cost-of-living adjustments.

Slope

The slope of a linear function represents the rate of change of the dependent variable relative to the independent variable. It is calculated as the difference in the values of the dependent variable divided by the difference in the values of the independent variable, providing a measure of how quickly the quantity being modeled is increasing or decreasing over time.

Transcript

00:01 So in this problem, we're looking at another linear application.

00:03 We're actually told in this case, right, that we're talking about things in terms of t, right, where t represents our time, the number of years since 1980.

00:14 And our output represents our cpi, our consumer price index.

00:19 So one of the things that we know is it was in 1983.

00:24 And again, we're talking about years since 1980.

00:26 So three years after 1980, the cpi was 100.

00:33 In 2008, so 2008, and to think about this, right, if we want to know how many years, it was 28 years from 1980, the index would be 215 .3.

00:50 Now we have two points, and considering that it's approximately linear, we can write an equation to represent this situation.

00:58 So we're going to look at the changes of the ys over the changes of the x's, which will give us our slower.

01:03 So in this case, i'm going to take 215 .3 and subtract 100, which is 115 .3.

01:10 Then i'm going to take 28 minus 3, which is 25.

01:14 So that's going to give us an approximate slope of 4 .612.

01:22 So y equals 4 .612x plus b.

01:28 I'm going to use the point here.