Fisher Price Index Definition
The Fisher Index is a consumer price index used to measure the increase in prices of goods and services over a period of time and is calculated as the geometric mean of the Laspeyres Index and the Paasche Price Index.
Fisher Index Formula
where,
LPI = Laspeyres Price Index = ∑(Pn,t) * (Qn,0) * 100 / (Pn,0) * (Qn,0)
PPI = Paasche Price Index = ∑(Pn,t) * (Qn,t) * 100 / (Pn,0) * (Qn,0) ,
where
- Pn,t is the price of the item at the n^{th} period
- Pn,0 is the price of the item at the base period
- Qn,t is the quantity of the item at the n^{th} period
- Qi,0 is the quantity of the item at the base period
Examples of Fisher-Price Index
Below are the examples of a fisher price index.
Example #1
Let us find the Fisher-price Index for three items whose price and quantity sold are given for three years. For the current year designated as Year 0 the prices in dollars and the quantity are given as follows:
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Year 0 | Year 1 | Year 2 | |||||||
A | B | C | A | B | C | A | B | C | |
Price | 20 | 10 | 15 | 22 | 11 | 26 | 24 | 12 | 28 |
Quantity | 15 | 20 | 25 | 20 | 20 | 17 | 25 | 20 | 15 |
Value | 300 | 200 | 375 | 440 | 220 | 442 | 600 | 240 | 420 |
First, we will calculate the Fisher-price Index for Year 0 using Laspeyres Price Index and Paasche Price Index.
Laspeyres Price Index for Year 0 –
- For Year 0 the Laspeyres Price Index (LPI) = (20*15+10*20+15*25)*100/ (20*15+10*20+15*25)
- = 100
Paasche Price Index –
- Paasche Price Index = (20*15+10*20+15*25)*100/ (20*15+10*20+15*25)
- = 100
Fisher Price Index for Year 0 –
- Fisher Index(FPI) = (100*100)^0.5
- = 100
Similarly, we find the indexes for Year 1 and 2 as given.
For Year 1
Laspeyres Price Index
- LPI = (22*15+11*20+26*25)*100/ (20*15+10*20+15*25)
- = 137.14
Paasche Price Index
- PPI = (22*20+11*20+26*17)*100/ (20*15+10*20+15*25)
- = 125.94
Fisher Index (FPI)
- FPI = (137.4*125.94)^0.5
- = 131.42
For Year 2
Laspeyres Price Index
- LPI = (24*15+12*20+8*25)*100/ (20*15+10*20+15*25)
- = 148.57
Paasche Price Index
- PPI = (24*12+12*20+28*15)*100/ (20*15+10*20+15*25)
- = 144
Fisher Index
- FPI = (148.57*144)^0.5
- = 146.27
We have given a tabular representation of the indexes in the following table.
Example #2
Let us take the case of three very commonly used fuels: petrol, diesel, and kerosene and calculate the price indices for three years.
The price in dollars and quantities in liters are shown in the following table.
Year 0 | Year 1 | Year 2 | |||||||
Petrol | Diesel | Kerosene | Petrol | Diesel | Kerosene | Petrol | Diesel | Kerosene | |
Price | 60 | 70 | 50 | 65 | 78 | 52 | 50 | 65 | 45 |
Quantity | 10 | 10 | 15 | 20 | 15 | 18 | 8 | 5 | 10 |
Value | 600 | 700 | 750 | 1300 | 1170 | 936 | 400 | 325 | 450 |
We can see that the price of fuels increased in Year 1 and decreased in Year 2. Did you notice that the quantities also show a similar trend, which is not surprising as we know that oil and gas exploration companies often reduce production when the price of crude oil (the raw material) fall?
The table showing the values of the indices, in this case, is given below and can be derived exactly in the same manner as shown in the above example.
Advantages of the FPI
- FPI is often called the real index as it corrects for the upward bias of the Laspeyres Price Index and the downward bias of the Paasche Price Index by taking the geometric average of the two weighted indices. It uses both current year and base year quantities as weight.
- Although it is not very frequently used index owing to its structural complexity and the number of variables required, it has very widespread use in academic circles and research.
Disadvantages of the FPI
- The only limitation of the FPI is it is a little more complex constructs than the other two.
- The quantities of the future years have to be forecasted while in the case of Laspeyres Price Index only the future prices have to be found out.
Conclusion
Although the Fisher Index is the better of the three indices Laspeyres Price Index is more commonly used for inflation calculations. But if we can make an accurate forecast of future quantities of an item Fisher-price index gives a more accurate measure.
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