In the audio session, two methods of constructing a price index for bread were described. They were called the ‘previous year? method and the ‘base year? method. In both cases, the value of the index in the base year is 100. So, for the base year method,
For the previous method,
Both methods lead to the same values for the price index (except for possible rounding errors).
For a single item, like the bread in the audio session, the price index gives a series of numbers that show how the price of the item changed over time. But the yearly prices for the item also show how its price changed over time, and calculating a price index does not help a great deal in understanding the price changes.
However, a price index can also be calculated for a whole basket of goods, and in this context, the price index is often very helpful in understanding the price changes. Either method (previous year or base year) could be used to construct a price index for the shopping basket of five items in Table 6. Let's use the ‘previous year? method here because it will be more useful later. The value of the index in the base year is 100. Then each year the value of the index for the previous year is multiplied by a price ratio for the basket of goods for the year to obtain the new value of the index.
The basket price ratio is the weighted mean of the price ratios for the five items: the weights used are proportional to the amounts spent on each item in a typical week. Recall from Subsection 3.2 that it is only the relative size of the weights that determines their effect, not their absolute value. So as long as the relative size of the amounts spent on the items remains unchanged from year to year, there is no need to change the weights. For simplicity, assume that the various weights remained unchanged between 1997 and 2004. Table 17 shows the price ratios for the five items for January 1998 relative to January 1997 and the weights used.
| Item | Price ratio | Weight (average 1997 weekly bill in pence) |
|---|---|---|
| Large loaf | 0.927 | 220 |
| Milk | 0.972 | 420 |
| Eggs | 0.969 | 50 |
| Potatoes | 2.000 | 55 |
| Sugar | 0.947 | 25 |
Find the price ratio for the basket for January 1998 relative to January 1997 by finding the weighted mean of the five price ratios in Table 17.
If the price ratios are called r and the weights w then Σrw = 794.305 and Σ = 770. So the weighted mean of the price ratios is
So the price ratio for the basket for January 1998 relative to January 1997 was 1.032.
Calculations similar to the one in this activity resulted in the year-on-year basket price ratios shown in Table 18. (For instance, the one-year price ratio for 1999 is the weighted mean of the price ratios for January 1999 relative to January 1998.)
| Year | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 |
| Price ratio | 1.032 | 1.041 | 0.963 | 1.099 | 1.043 | 1.016 | 1.034 |
Taking 1997 as the base year, the value of the basket price index (hereafter simply called the index) is set to 100 in January 1997. The value of the index in January 1998 is then:
As was the case for the index for bread, each year the value of the index is found by multiplying the value of the index a year earlier by the price ratio for the year. So the value of the index for January 1999 is given by the following computation.
Using the method that was just used to calculate the value of the index for 1999, calculate the value of the price index for the shopping basket in Table 17 for each of the years 2000 to 2004, using the price ratios in Table 18.
The values of the price index are given in the table below.
| Year | Value of index in January |
|---|---|
| 2000 | 107.4 × 0.963 = 103.4 (approx) |
| 2001 | 103.4 × 1.099 = 113.6 (approx) |
| 2002 | 113.6 × 1.043 = 118.5 (approx) |
| 2003 | 118.5 × 1.016 = 120.4 (approx) |
| 2004 | 120.4 × 1.034 = 124.5 (approx) |
Thinking in terms of the base year method of calculation, the price index for a given year is simply the price ratio for that year relative to the base year, multiplied by 100. The value of the price index was 103.4 in January 2000, so the basket cost 1.034 times as much in January 2000 as in January 1997. That is, the price of the basket had risen by 3.4%. The percentage increase in the price of the basket between any two years can be found from a price ratio, calculated as the value of the index in the later year divided by the value in the earlier year. This price ratio is then used to find the percentage price rise. For example, the price ratio for January 2001 relative to January 1998 is worked out as follows.
So the price of the basket rose by 0.101 as a proportion, or 10.1%, between January 1998 and January 2001.
(a) Find the proportional rise and the percentage rise in the price of the shopping basket between January 2000 and January 2004.
(b) Find the proportional rise and the percentage rise in the price of the shopping basket between January 1999 and January 2002.
(a) The price ratio for January 2004 relative to January 2000 is given by
So the price of the basket rose by about 20.4%, or 0.204 as a proportional rise, between January 2000 and January 2004.
(b) The price ratio for January 2002 relative to January 1999 is given by
So the price of the basket rose by about 0.103 as a proportional rise, or 10.3%, between January 1999 and January 2002.
The basket price ratio was found by weighting the price ratios of the five items according to how much was spent on them in a typical week. The relative amounts spent each week on the items in the basket were assumed to remain unchanged over the years. However, this is not always a reasonable assumption. For example, over a number of years, less may be spent on milk and more on eggs. This information could be incorporated into the calculation of the index by changing the weights used to calculate the basket price ratio each year to reflect changes in spending. This is what happens for the official price indices published by the UK Government. This is an advantage of building a price index by calculating one-year price ratios and the previous year method over the base year method: the previous year method allows changes in spending patterns to be taken into account more easily.
Of course, this basket of five items is not really very representative of all the expenditure made in a typical week. Many more goods and services are taken into account by the government in the calculation of the price indices.
This would be a good time to check that you have a record of any terms or ideas. Check your record entries with the information in the summary box below. The main points about the construction of a price index whose value is calculated once a year are summarised in the box. ‘Price index? might be a good one.
A price index
The starting value of the index (that is, in the base year) is 100. Each year, the new value of the index is calculated according to the previous year method, by using the following formula.
Using a price index
The percentage price increase between any two years covered by an index can be found by first finding the price ratio for the later year relative to the earlier year:
Then the percentage price increase is given by this final formula.
After studying this section, you should be able to:
♦ use a weighted mean to find an average price ratio (Activities 19 and 20);
♦ convert a proportional or percentage price increase into a price ratio or index and vice versa (Activities 18 and 19);
♦ calculate a proportional or percentage price increase and a price ratio from a price index (Frame 6, Activity 22);
♦ use the base year and previous year methods to calculate a price index (Frames 4 and 9, Activity 21).
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