Price Indexes: Part 1

1. Price Indexes:Part 1 Marc Prud’Homme University of Ottawa Last update: 14/09/12

2. Definition • Index numbers : Numbers that measure a variable’s magnitude at one time or place relative to the same variable’s magnitude at another time or place. • Temporal index numbers arecomparisons between one time and another. • Spatial index numbers are comparisons between one geographic location and another.

3. Definition (cont’d) • The variable in question can be an individual • Price • Quantity • Value figure • More often it is entire sets of these prices, quantities, or values that are being compared. • When single items are compared rather then sets of them, index construction is simple. • When we are comparing the prices or quantities of many items, things quickly become more complicated.

4. Definition (cont’d) • This part of the course discusses the construction of index numbers and also introduces the major types of business and economic index numbers that are regularly published and used in Canada and elsewhere. • Various methods are used to construct price indexes, which are numbers that measure the level of a single price or a set of prices at one time or place relative to another time or place. • Simple price indexes • Un-weighted composite price indexes • Weighted aggregative price indexes

5. Simple price indexes • A simple price index or price-relative index is a number that compares the price of a single item at one time or place with the price of the same item at another time or place.

6. Simple price indexes • The Table on the forthcoming slide illustrates the computation of a simple temporal price index series. • We have data on the average price of grade-A eggs in Ottawa during the year.

7. Simple price indexes • Calculate a simple price index in two steps: • Relate the price of each period to the price in an arbitrarily chosen base period or reference period. • Convert the price relatives in column 3 to the simple price indexes of column 4. • In doing so, we follow tradition and express the base-year index as 100. • The formula summarizes the procedure: • Confirm the 1999 price index:

8. Simple price indexes

9. An interspatial price index • Instead of having years in column 1 of the previous Table, you will have geographical areas.

10. Un-weighted composite price indexes • Typically, economic decision makers are concerned with comparisons that are considerably more complex than those discussed so far. • They need to compare entire sets of prices prevailing at a given place at one time with the corresponding set of prices prevailing at the same place at a different time. • Several possible approaches to such a calculation exist. • We begin with the construction of un-weighted composite price indexes along with its flaws.

11. Un-weighted composite price indexes • A composite price index is a number that compares a set of prices for a variety of items at one time or place with such a set for the same items at another time or place.

12. An un-weighted average of simple price indexes • One way to construct a composite price index that combines the possibly divergent price movements of different items in a single number is to compute an un-weighted average of the separate simple price indexes for all the items in question. • Such an average is constructed on the (often unrealistic) assumption that all items are equally important.

13. Example

14. An un-weighted average of simple price indexes • Example: Using the Table from the next slide, compute a composite price index for breakfast food prices for 2001.

15. Un-weighted composite price indexes • This method of price indexation was employed for the first time in 1764 by Giovanni R. Carli. • He set out to measure the effect on European prices of the influx of precious metals following the discovery of the New World. • He applied this formula to the prices of grain, oil, and wine in 1500 and 1750.

16. Un-weighted composite price indexes • An altogether different approach to constructing a composite price index does not require the prior computation of individual price relatives at all. • You can also compute the average price of all relevant products in the year for which the index is desired and relating that average to the similar average of all base year prices.

17. Example

18. Example • Calculating the index where different quantity units are used (Part B) yields: • These results are disturbing! • Furthermore, they are different from the Carli index.

19. Un-weighted composite price indexes • The method of price indexation was employed by Nicolas Dutot in 1738. • Just for completeness, lets apply the Carli index to the data from part B. • Note that the Carli index is not sensitive to changes in the unit of measurement.

20. Weighted aggregative price indexes • The Carliand Dutot indexes had one major flaw. • They treat all goods as equally important. • Weighted aggregative price indexes remedy this problem. • Numerous weighted aggregative price indexes exist. • Some are more well known than others. • Each of them computes a sum of quantity-weighted prices for the current year and then relates that sum to a similar sum for the base year. • The indexes differ from one another with respect to the way in which the quantity weights are chosen.

21. The Laspeyres price indexes • The Laspeyres price index is a weighted aggregative price index that uses base-quantity weights, Q0, in both the numerator and denominator of the price index formula. • It looks like this: • This is a composite (temporal) price index

22. The Laspeyres price indexes • With the Laspeyres formula, the objections to the un-weighted aggregative index are eliminated. • The use of differentiated quantity weights eliminates the equal treatment of all goods. It allows us to take account of the importance of each product relative to the entire set of products. • The fact that each price is multiplied by the quantities makes the choice of quantity units unproblematic because the quantity units cancel out.

23. The Laspeyres price indexes • Thus a price of $1 per dozen eggs, when multiplied by a quantity of 100 dozens yields $100. • An equivalent price of $83.33 per thousand eggs when multiplied by the equivalent quantity of 1.2 thousand eggs, also yields $100.

24. Example

25. The Laspeyres price indexes • Example calculation • According to the Laspeyres formula, breakfast food items in 2001 were, on average, 6.7% more expensive than in 2000.

26. The Paasche price indexes • The Paasche price index is a weighted aggregative price index that used current-year quantity weights, Qt, in both the numerator and denominator of the price index formula. • It looks like this: • This is a composite (temporal) price index

27. Example

28. The Paasche price indexes • Example calculation • According to the Paasche formula, breakfast food items in 2001 were, on average, 4.5% more expensive than in 2000.

29. The index-number problem • Note how the outcomes differ between the Laspeyres and Paasche price indexes: • 106.7 vs. 104.5 • Odd given that the indexes attempt to measure the same thing: price level changes between 2000 and 2001. • However, they actually measure two different things.

30. The index-number problem • Laspeyres tells us how more or less money people would have had to spend in 2001, compared to 2000, if they had tried to buy in both years the physical quantities that they in fact bought in 2000. • In our example: 6.7% more • Paasche tells us how more or less money people would have had to spend in 2001, compared to 2000, if they had tried to buy in both years the physical quantities that they in fact bought in 2001. • In our example: 4.5% more • Both of these stories are pure fiction.

31. The index-number problem • Each serves the sole purpose of constructing two value figures (found in the numerator and the denominator of the respective formulas) that: • Conveniently contain the same quantity component • Can, therefore, differ from one another only as a result of temporal differences in prices • Can, thus, be used to estimate price change • Yet, in truth, people bought year 2000 quantities in 2000 and different year 2001 quantities in year 2001.

32. The index-number problem • What then is the true price level change between these two years? • Sadly, the question cannot be answered because, when comparing the 2001 price set with the 2000 price set, it is just as logical to use the 2000 quantity weights (Laspeyres formula) as it is to use the 2001 quantity weights (Paasche formula). • In fact, one can argue that sometimes it can even be better to use 1980 quantity weights. • Brings us to the index-number problem.

33. The index-number problem • The index-number problem is an insoluble problem, which refers to the fact that it is often possible to construct different indexes (with different outcomes) for a given phenomenon, depending on which one of several equally logical sets of weights is applied to individual index components.

34. The index-number problem • The discrepancy between the Laspeyres and the Paasche indexes will likely widen as the distance increases between the base period and current period. • If we tried to measure price changes between 1900 and 2000… • People’s tastes change • Consumer change their buying patterns because of changes in relative prices. • Income’s change • As a result of these and other factors, a year 1900 quantity set (what Laspeyres would use) would be quite different from the year 2000 quantity set (what Paasche would use). • The indexes would differ.

35. Other weighted aggregative price indexes • The Marshall-Edgeworth price index combines features of the Laspeyres and Paasche by using the sum of the base-year and current year quantities (Q0 + Qt) as weights.

36. Other weighted aggregative price indexes • The Walsh price index combines features of the Laspeyres and Paasche by using the square root of the base-year and current year quantities as weights.

37. Other weighted aggregative price indexes • The Tornqvist price index is the geometric average of the price relative of the current to base period prices weighted by the arithmetic average of the value shares for the two periods. • The denominators in the exponent are the sums of total expenditure in each of the two periods.

38. Superlative indices • The previous three indexes (Walsh, M-E, Tornqvist) are superlative price indexes. • Superlative price indexes treat prices and quantities equally across periods. • They are symmetrical and provide close approximations of cost of living indices and other theoretical indices used to provide guidelines for constructing price indices.

39. Superlative indices • All superlative indices produce similar results and are generally the favoured formulas for calculating price indices. • A superlative index is defined technically as an index that is “exact” for a flexible functional form that can provide a second-order approximation to other twice-differentiable functions around the same point.

40. Fisher’s quality tests and his ideal index • With so many formulas to chose from, the index maker naturally looks for guidance to facilitate the inevitable choice. • We may well ask: What would make one index number better than another? • In his classic work, The making of index numbers, first published in 1922, the American economist Irving Fisher set out some principles so that superior index numbers can be distinguished from inferior ones. • He developed index-number quality tests for which we shall cover three here: • Time reversal test • Factor-reversal test • Circularity test

41. The time-reversal test • Consider the price of a single item, such as a loaf of bread. If the loaf costs $1 in year 0 and $2 in year 1, we can describe the matter in two ways: • We can say that the year 1 price is double the year 0 price • We can say that the year 0 price is half the year 1 price • Either way, the story is the same. • Fisher’s position was that price index numbers, which combine the prices of many items, should be able to tell an equally consistent story, regardless if year 0 or year 1 are used as the basis of comparison. • A price-index formula passes Fisher’s time-reversal test if, in a comparison of prices at two dates, we get consistent (reciprocal) results regardless of which date we choose as the base.

42. The time-reversal test • For example, if the formula indicates the year 2000 prices are twice as high as 1990 prices one 1990 is used as a base, then it should also indicate that 1990 prices are half as high as 2000 prices when 2000 is used as a base. • Denoting the price index by PI and the two dates by 0 and 1, passing the test requires that • or that

43. The factor-reversal test • Now consider a price and quantity of a single item, such as bread. If a loaf cost $1and you buy 4 loafs, we can figure the value of your purchase as $1 x 4 = $4. • For individual items we take it for granted that price X quantity = value • Fisher argued that index numbers that combine the price or quantities or the values of many items (whichever the case may be) should indicate the same logical relationship. • An index of prices multiplied by an index of quantities should equal an index of values.

44. The factor-reversal test • Fisher’s factor reversal test presupposes that the weights used in a price index formula are quantity weights and a corresponding quantity index is derived by interchanging the Ps and Qs in the formula. • This being so, a price index formula passes the test if division of the price index by 100, and subsequent multiplication by a corresponding quantity index, produces a number that equals an independently derived value. • Denoting the price index by PI and the quantity index by QI, passing the test requires that

45. The circularity test • Consider once again the price of a single item, such as a loaf of bread. If the loaf costs one dollar in year zero, two dollars in year one, and four dollars in year two, we can describe the matter as follows: • we can say that the year 1 price is double the year 0 price • we can say that the year 2 price is double the year 1 price • we can say that the year 2 price is quadruple the year 0 price • Note that we could have derived the last statement without looking at the original year 0 and year 2 prices and by simply multiplying the key numbers found in the first two comparisons: • 2 X 2 = 4 • Fischer argued that index numbers that combine the prices of many items should indicate the same logical relationship.

46. The circularity test • Accordingly, a price index formula passes Fisher’s circularity test if the price index of year 1 with year 0 base, when multiplied by the price index of year 2 with year 1 base (and divided by 100), equals the independently calculated price index of year 2 with year 0 base. • For example, if the formula indicates that prices have doubled between 1990 and 1995 and then have doubled again between 1995 and 2000, it should also indicate that prices have quadrupled between 1990 and 2000, when the prices of the latter two years are compared directly. • Thus, passing the test requires that:

47. Testing the Laspeyres Index • All of Fisher’s quality tests make eminent sense. • Surely, if A is twice as big as B, an examination of B should reveal it as being half as big as A (the time-reversal test). • Surely, if A times B equals C, the aggregation of lots of A’s times the aggregation of lots of B’s should equal the aggregation of lots of C’s (the factor-reversal test). • Surely, if B is twice A and C is twice B, C should be found to equal four times A (the circularity test). • Yet, surprisingly, many index numbers in common use fail to meet these tests! • The following calculations illustrate this sad truth in the case of the Laspeyres index.

48. Testing the Laspeyres Index The time-reversal test • From previously: • Now calculate a Laspeyres price index for 2000 with 2001 base as • The Laspeyres price index does not pass the time-reversal test because

49. Testing the Laspeyres Index The factor-reversal test • From previously: • Now calculate a Laspeyres quantity index for 2000 with 2001 base as • The corresponding value index, VI, is found to equal

50. Testing the Laspeyres Index The factor-reversal test • The result indicates that 2001expenditures exceeded 2000 expenditures by 34.4%. • The Laspeyres index does not pass the factor-reversal test because