Weighted Indices
A price relative is an index number showing changes in a single price  at different times. It is usually of interest to find an index number representing changes in a set of prices rather than just one. Suppose in 1996 a company buys 20 cars and 30 lorries, and suppose the price of a lorry rises from £100,000 to £120,000. In that case the price relative of a lorry is 120% and that of a car is 110%, but neither figure on its own tells us how the company’s actual costs have changed. However we can divide the company’s total costs in 1997 by its total costs in 1996:
Percentage change in costs = = =119.37%
In general we can use this idea to show how the cost of a given basket of goods has risen; we multiply the quantity of each good by its price in each of two years to get its total cost in year 1 and year 0, and then divide the year 1 total price by the year 0 total price. This is called a weighted price index number for period 1. Generalising a bit, the index for year t is:
	 
where the w is the weight representing the amount of the item in the basket. 
Laspeyre and Paasche indexes
An unfortunate complication is the following: supposing in 1997, instead of buying 20 cars and 30 lorries like in 1996, the company changes the size of its basket of consumption and buys 15 cars and 31 lorries. In that case there are two possible measures of the rise in the cost of the company’s purchases. We might either ask how much it would have to pay in 1997, in order to buy the same things that it actually purchased in 1996; or we might ask how much it would have to pay in 1996, in order to buy the same things it actually purchased in 1997. In each case we would divide the 1997 cost by the 1996 cost, but the result would be a different index because the weights (w) are in one case taken from 1996 and in the other from 1997. This means we have no general measure of inflation and is a part of what economics calls the index number problem. 
The two types of indices are named after two people who were involved in developing them. One is called the Laspeyres, and the other the Paasche index. If we use the letters p and q to stand, as usual, for price and quantity, then the Laspeyres price index for date t with base year 0 is weighted by the quantities of each item at the base date (date 0), and is as follows
	 
Thus the Laspeyres index uses the weights of the base year. Year t is usually termed the current year, and we construct a Paasche index with the weights of the current year. Sometimes it is called a current weighted index. The  Paasche price index is
	
A Laspeyres price index for period t shows the extent of price changes since period 0 on the assumption that the expenditure pattern was the same in period t as in period 0. A Paasche index shows the change assuming the expenditure pattern was the same in period 0 as in period t. If  a price relative or a price index is 130 for period t; this is interpreted as follows: “prices have increased by 30 percent from period 0 to period t”.