(each quantity of an item is multiplied by the same factor of either , for the first period, or , for the later period) then the index value will be one. Proportionality test: If each price in the original period increases by a factor α then the index should increase by the factor α. Invariance to changes in scale test: The price index should not change if the prices in both periods are increased by a factor and the quantities in both periods are increased by another factor. In other words, the magnitude of the values of quantities and prices should not affect the price index. Commensurability test: The index should not be affected by the choice of units used to measure prices and quantities. Symmetric treatment of time (or, in parity measures, symmetric treatment of place): Reversing the order of the time periods should produce a reciprocal index value. If the index is calculated from the most recent time period to the earlier time period, it should be the reciprocal of the index found going from the earlier period to the more recent. Symmetric treatment of commodities: All commodities should have a symmetric effect on the index. Different permutations of the same set of vectors should not change the index. Monotonicity test: A price index for lower later prices should be lower than a price index with higher later period prices. Mean value test: The overall price relative implied by the price index should be between the smallest and largest price relatives for all commodities. Circularity test: Given three ordered periods , , , the price index for periods and times the price index for periods and should be equivalent to the price index for periods and . Quality change Price indices often capture changes in price and quantities for goods and services, but they often fail to account for variation in the quality of goods and services. This could be overcome if the principal method for relating price and quality, namely hedonic regression, could be reversed. Then quality change could be calculated from price. Instead, statistical agencies generally use matched-model price indices, where one model of a particular good is priced at the same store at regular time intervals. The matched-model method becomes problematic when statistical agencies try to use this method on goods and services with rapid turnover in quality features. For instance, computers rapidly improve and