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.