Question

The total price of purchasing a basket of goods in the United Kingdom over four years is: year \(1=\mathrm{f} 940\) year \(2=\mathrm{f} 970,\) year \(3=\mathrm{f} 1000,\) and year \(4=\mathrm{E} 1070\) Calculate two price indices, one using year 1 as the base year (set equal to 100 ) and the other using year 4 as the base year (set equal to 100 ). Then, calculate the inflation rate based on the first price index. If you had used the other price index, would you get a different inflation rate? If you are unsure, do the calculation and find out.

Short answer

The price indices using year 1 and year 4 as base years are: \( Price\,Index\,(Year\,1\,Base) = [100, 103.19, 106.38, 113.83] \) \( Price\,Index\,(Year\,4\,Base) = [87.85, 90.65, 93.46, 100] \) The inflation rate from year 1 to year 4 is 13.83%, and it remains the same regardless of the base year used for the price index calculation.

Step-by-step solution

Calculate the price index with year 1 as the base year

To calculate the price index of a year using year 1 as the base year, we need to follow this formula: \( Price\,Index\,for\,year\,n = \frac{Price\,of\,goods\,in\,year\,n}{Price\,of\,goods\,in\,year\,1} * 100 \) Calculating the price index for each year: Price Index for year 1: \( Price\,Index\,year\,1 = \frac{940}{940} * 100 = 100 \) Price Index for year 2: \( Price\,Index\,year\,2 = \frac{970}{940} * 100 ≈ 103.19 \) Price Index for year 3: \( Price\,Index\,year\,3 = \frac{1000}{940} * 100 ≈ 106.38 \) Price Index for year 4: \( Price\,Index\,year\,4 = \frac{1070}{940} * 100 ≈ 113.83 \)

Calculate the price index with year 4 as the base year

Now, we will calculate the price index using year 4 as the base year: \( Price\,Index\,for\,year\,n = \frac{Price\,of\,goods\,in\,year\,n}{Price\,of\,goods\,in\,year\,4} * 100 \) Calculating the price index for each year: Price Index for year 1: \( Price\,Index\,year\,1 = \frac{940}{1070} * 100 ≈ 87.85 \) Price Index for year 2: \( Price\,Index\,year\,2 = \frac{970}{1070} * 100 ≈ 90.65 \) Price Index for year 3: \( Price\,Index\,year\,3 = \frac{1000}{1070} * 100 ≈ 93.46 \) Price Index for year 4: \( Price\,Index\,year\,4 = \frac{1070}{1070} * 100 = 100 \)

Calculate the inflation rate based on the first price index

The inflation rate is calculated as the percentage change in the price index from one year to another. We can use the following formula: \( Inflation\,Rate\,between\,year\,n1\,and\,n2 = \frac{Price\,Index\,year\,n2 - Price\,Index\,year\,n1}{Price\,Index\,year\,n1} * 100 \) Calculating the inflation rate from year 1 to year 4: \( Inflation\,Rate\,1\,to\,4 = \frac{113.83 - 100}{100} * 100 = 13.83\% \)

Analyze if the inflation rate would be different using the other price index

We can observe that the relative change in the price indices from year 1 to year 2, from year 2 to year 3, and from year 3 to year 4 is the same, regardless of the base year used for the calculation. Their relation will hold: \( \frac{Price\,Index\,year\,n (Base\,Year\,1)}{Price\,Index\,year\,n (Base\,Year\,4)} = \frac{Price\,Index\,year\,1 (Base\,Year\,1)}{Price\,Index\,year\,1 (Base\,Year\,4)} \) Thus, calculating inflation based on the second price index would yield the same result as calculating inflation based on the first price index: In conclusion, the price index using year 1 and year 4 as the base year is: \( Price\,Index\,(Year\,1\,Base) = [100, 103.19, 106.38, 113.83] \) \( Price\,Index\,(Year\,4\,Base) = [87.85, 90.65, 93.46, 100] \) And the inflation rate from year 1 to year 4 is 13.83%, regardless of the base year of the price index we use.

Key concepts

Inflation Rate Calculation

Inflation is a term that defines how prices rise over time, and calculating it can give us valuable insight into economic trends. The inflation rate calculates how much the price level of goods and services has increased over a specific period, and it's usually expressed as a percentage. To calculate inflation, we need a reliable price index for the start and end of the period.
The formula to calculate the inflation rate between two periods is:

• Subtract the price index of the initial period from the price index of the later period.
• Then, divide this difference by the initial period's price index.
• Finally, multiply the result by 100 to convert it into a percentage.

This tells us how much, in terms of percentage, the price of goods and services increased relative to our starting point. Understanding this concept helps us grasp the broader economic shifts that influence everything from daily expenses to global financial markets.

Base Year

In economics, the base year is a pivotal reference point for comparing the value of different years in a time series. When calculating price indices, the base year is typically assigned an index value of 100, making it a benchmark against which other years are compared.
Using the base year allows us to see how prices have changed over time by creating a "zero point" from which future price movements can be measured. This simplicity makes it easier for anyone to quickly understand whether the prices have increased or decreased relative to the base year.
It's important to note that choosing a different base year will not alter the percentage changes in inflation rates, but it might change the numerical values of the index. Despite this numerical shift, the relative comparison remains constant.

Basket of Goods

The term "basket of goods" is used in economics to represent a set collection of goods and services that are purchased by average households. It acts as a mirror reflecting consumption patterns and serves as a crucial component of the Consumer Price Index (CPI).
The basket includes everyday items such as food, clothing, healthcare, and housing. By keeping track of the total cost of this basket over time, economists can measure how the price level, and hence inflation, is progressing.

• The total cost of the basket in the base year provides the standard reference point.
• Then, subsequent years' costs are compared to this base to assess price changes.

Changes in the basket or its prices over the years illustrate shifts in consumer habits and cost of living adjustments. This concept allows households, businesses, and governments to make informed financial and policy decisions.