Have you ever heard someone say, "Money just isn't worth much anymore?" There is some truth to it. As inflation increases the cost of goods and services, it takes more money to buy the same amount. That is one reason some employers give cost-of-living increases. Economists use index numbers to calculate such changes over a specified period in time.
Calculating the Percentage of Price Changes
This is one of the simpler index number calculations. By comparing a price at one time to the price for the same item later in time, the percentage of change is calculated. For example, suppose a fountain drink at the local diner cost $.25 in 1952, but the same drink costs $1.50 today. The index number is the new cost ($1.50), the base period is 1952, and the variable for that base is the old cost ($.25). The variable -- the number compared against -- gets a value of 100. The percentage of change is calculated as follows:
The result shows that since 1952, the cost of a fountain drink at the local diner has risen 600%. When using an index number to show the price change for a single item, it is a price relative. The Retail Price Index is a term most people have heard. It measures price changes for a wide variety of products for the same period and gives a measurement of the cost of living.
By assigning "po" to the older price and "pn" to the newer price, we can show the price relative as (po/pn)*100.
Index numbers do not carry quantities. In the example above, the index in 1952 is 100, and in 2009 is 600.
The Expenditure Index
Calculating changes in costs for multiple items over time works much the same way. The big difference is that now there is more than one cost to include in each calculation. For example, suppose your company had a picnic for employees in 2004 that was a huge success, and there has been one every year since. This year, however, economic times are not as good, and management has dictated that if the picnic's cost rises more than 50 percent over 2004's prices, the picnic is off. In that five-year period, the number of employees has increased from 50 to 75. Using some basic items as comparison, we can build a chart like the one below to make it easier to see our items and prices. We can leave out the currency sign because index calculations are independent of currency.
The total cost of each picnic is calculated. The Greek uppercase letter S (sigma or ') means "the sum of all that is before," so the cost in 2004 is:
In 2009, the estimated cost is:
By adding up the costs for each picnic, we are able to treat them as single items, so we may use the same price relative calculation used for the fountain drink.
There would be a 50-percent increase in the picnic's cost from 2004 to 2009, taking into account the higher costs and number of employees (which meant buying larger quantities of each item). It looks like the picnic is on!
Weighting and Simple Aggregate Index
Notice that we multiplied the price and quantity of each item to get a total cost for the item. This is weighting. Had we used simply the base price of each item, we could calculate the simple aggregate index. This number would not have been useful for several reasons. Firstly, there are different quantities of the same item in each year. Secondly, the units of measure for items do not match: 12-packs for beer, 8-packs for buns and hot dogs, pounds for hamburger meat, and single bags of chips. As the saying goes, it would be like comparing apples and oranges.
The Base Weighted Index (Laspeyre's Index)
To correct that problem, we use the base weighted index. This index measures price changes from a base year, in our case 2004. It gets its name from the fact that we use the quantities bought in the base year to weight the unit prices in both years. Any change in the calculation is then due to nothing but price changes since we have kept the quantities constant.
Using the same picnic chart, we have:
Exactly what does this tell us? It means the real increase in cost is 67.38 percent from 2004 to 2009 (not the original 50 percent we calculated), because our new calculation weighted the quantities, thereby making them constant from one period to the next. We are now working strictly with changes in price, and it look there won't be a picnic after all.
One of the primary reasons for this method is because when making complex calculations like the ones needed to find cost of living increases, quantities are irrelevant in many cases. Although we may be able to by a quantity of apples, how would we quantify a public service such transportation?
One of the problems with doing index calculations is choosing a suitable base period. Care is necessary in choosing a period in which unnatural fluctuations did not occur. For example, an extremely hot summer with little rain might devastate a certain crop, thereby raising the price artificially because there is so little volume available. Another example is extreme changes in fashion that cause a certain style to sell poorly while another skyrockets. Sometimes a chain-based system can avoid this problem, where each year becomes the base period for the following period, rather than having several or many years between the base and the current periods. This is helpful for showing period-to-period comparisons.