Index Numbers: Laspeyres, Paasche, and Fisher



Index Numbers: Laspeyres, Paasche, and Fisher Price and Quantity Indexes

In defining an index number, always start with a ratio of sums. Each component of each sum is a price times a quantity. The number [pic] of such price/quantity products is equal to the number of goods or services included in the index.

[pic]

The next step is to insert dates for the “?” marks.

If you want a Price Index, make sure only the prices in the numerator and denominator correspond to different dates. Quantities have the same date: earlier in both numerator and denominator, for a Laspeyres Index; later, for a Paasche Index. Therefore:

Laspeyres Price Index: [pic] Paasche Price Index: [pic]

If you want a Quantity Index, make sure only the quantities in the numerator and denominator correspond to different time periods. Prices have the same date: earlier in both numerator and denominator, for a Laspeyres Index; later, for a Paasche Index.

Laspeyres Quantity Index: [pic] Paasche Quantity Index: [pic]

In general, Laspeyres and Paasche Price Indexes or Inflation Factors will differ. Their geometric averages are called Fisher Indexes (after Irving Fisher).

[pic]

[pic]

As a measure of the inflation factor (one plus the inflation rate), a Fisher Price Index is the square root of the product of Laspeyres and Paasche Price Indexes. As a measure of the growth factor (one plus the rate of growth), a Fisher Quantity Index is the square root of the product of Laspeyres and Paasche Quantity Indexes.

Question: Why are Fisher Indexes useful averages of Laspeyres and Paasche Indexes?

Answer: The advantage of the geometric averages is revealed by considering the ratio of nominal values where $G is the rate of change of nominal GDP:

[pic]

Note that both prices and quantities are different, comparing the numerator and the denominator. If the ratio were 2, for example, value has doubled ([pic]). A doubling of nominal value might be the result of price increases (inflation) or quantity increases (real growth) or some mixture of inflation and growth. How can the ratio of nominal values be split exactly into an inflation factor (where [pic] is the inflation rate) times a growth factor (where [pic] is the growth rate)?

[pic]

The split will be exact if the Fisher Price Index is used for the inflation factor and the Fisher Quantity Index is used for the growth factor. Just do the substitution to confirm this claim.

Square Root of the Product of a Fisher Price Index times a Fisher Quantity Index

= [pic]

Bring all terms under a single square root sign, cancel sums where possible and take the square root. The result is just the ratio of nominal values in adjacent time periods.

Conclusion: Fisher Index Numbers for the inflation and growth factors are a mathematically coherent way to average Laspeyres and Paasche Index numbers because these averages succeed in splitting the change in nominal value exactly into a change in prices and a change in quantities.

Example (three periods): Assume there are only two goods in the calculation of GDP, with prices and quantities as follows. Three periods of data are available.

In the first period, let each price and quantity be equal to one, so that nominal GDP is [pic].

Assume that the price of the first good rises for 50% to 1.5 in the second period, and then a further 33% in the third period, i.e. from 1 to 1.5 to 2.0. Note that the second change of 0.5 if a smaller percentage increase because the price has already gone up. Suppose the quantity of the first good increases to 2.0 in the second period (100% increase), and then remains at 2.0 in the third period (0% change).

Assume that the price of the second good rise from 1 to 1.2 (20% higher) and then to 1.5 (a further 25%) in the second and third periods, respectively; while the quantity of the second good increase to 3.0 (200% higher) and then to 4 (a further 33%).

[pic] [pic]

With these numbers you should be able to verify and complete the following table of sums. Prices change as you shift right from one column to the next; quantities change as you shift down from one row to the next. Nominal GDP in each period is therefore read off along the downward sloping diagonal:

[pic]

The first row holds quantities fixed at their period 1 levels. Therefore, 2.7/2.0 = 1.35 is the Laspeyres price index or inflation factor, using the first two periods which means that the “first” period is period 1. Prices have increased by 1.35 – 1 = 35% according to this measure.

The second row holds quantities fixed at the period 2 levels. Therefore, 6.6/5.0 is the Paasche price index or inflation factor, using the first two periods. Prices have increased by 1.32 – 1 = 32% according to this measure. The two measures differ because different quantities are used in the calculation.

Now use the first and second columns, which hold prices fixed at their period 1 and period 2 levels. Laspeyres and Paasche quantity indexes or growth factors, comparing the first two periods, are 5.0/2.0 = 2.5 and 6.6/2.7 = 2.44, which yields growth rates of 150% and 144.4%. Remember that to go from a “inflation factor” or a “growth factor” to a rate of inflation or a rate of growth, just subtract 1.

Sticking with the first two periods, the Fisher Price Index is the geometric average [pic] for an inflation rate of 33.49%. Similarly, the geometric average [pic] is the Fisher quantity index, indicating real growth of about 147%. Finally, multiply the Fisher Price Index times the Fisher Quantity Index. The result is the ratio of nominal values, 6.6/2 = 3.3 (apart from rounding error).

Repeat the exercise by calculating the Fisher price and quantity indexes using the second and third periods. This just means that the “first period” is period 2 and the “second period” is period 3.

The Selected NIPA Tables from include references to “chain weighted” indexes for levels of output and prices corresponding to different components of expenditure, such as Consumption and Investment. The chain index is just a connected series of Fisher Price indexes, or Fisher Quantity indexes. Each additional year of data corresponds to [pic] and what was new last year becomes old or ‘initial’ period [pic] data as time passes. This means that the corner elements [pic] and [pic] are not used in calculating chain indexes because the prices and quantities are separated by more than one period. They are used in calculating “fixed weight” index numbers such as the Consumer Price Index, which holds quantities constant for a number of years before updating them. It can be argued that fixed weight price indices, such as the CPI, overstate inflation by ignoring the fact that consumers will try to substitute goods that have risen in price by smaller percentages for goods that risen in price by larger percentages.

Calculations of Laspeyres, Paasche, and Fisher Index Numbers as measures of inflation factors and growth factors are straightforward once you have the above table of sums set up. Behind the calculations are more difficult conceptual issues, in particular:

(1) What goods should be included in the index?

(2) Are the goods really the same as time passes or do they change in quality and therefore usefulness? If the goods in the index are of better quality as time passes, the “true” price will increase less (or decrease more) than the actual price. This applies to computers and automobiles and many other high tech goods, but not to bread or milk. Price indexes for services such as medical care. A medical check up is far more expensive now that it was twenty years ago but more informative. Would you want to purchase an older, less informative check up at a lower price (if it were available)? So, was there inflation in the cost of medical care or not? It’s a hard question to answer.

Further note on price indexes.

A Laspeyres price index is equal to a share-weighted average of individual inflation factors looking forward (new price over old price) for each good, where the shares are for the initial period:

[pic]

Confirm this result by canceling a price in each term on the left and adding.

A Passche price index is equal to the reciprocal of a share-weighted average of individual deflation factors looking backward (old price over new price), where the shares are for the final period (say the next period):

[pic]

Confirm this result by canceling a price in each term in the denominator on the left, adding, and taking the reciprocal.

The main reason that economists think that the Laspeyres index for inflation will exceed the Paasche index is that shares are likely to change in a systematic way, rising for goods whose prices have increased the least and falling for goods whose prices have increased the most. This happens to the extent that consumers are willing to substitute away from goods that have gone up in price the most and towards goods that have gone up in price the least, including those whose prices have actually gone down.

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