 # Fisher Price Index

A consumer price index (CPI) used to measure the price development of goods and services over a given period of time.

The Fisher Price Index is a consumer price index (CPI) used to measure the price development of goods and services over a given period. The Fisher Price Index, also known as Fisher's Ideal Price Index, is a geometric average of two other indices: the Laspeyres Price Index and the Paasche Price Index.

Fisher's Ideal Price Index is referred to as the ideal index because it corrects the positive price bias of the Laspeyres Price Index and the negative price bias of the Paasche Price Index. Fisher's Ideal Price Index also satisfies both the time-reversal and factor reversal tests.

The time-reversal test is a test that requires that if the prices and quantities in two different periods are being compared as interchangeable, the resulting price index is the reciprocal of the original price index.

When an index meets this test requirement, the same result is derived regardless of the direction of change measured.

The direction of change can be measured forward in time from the first to the second period or backward from the second to the first period; the result will be the same.

The factor reversal test says that multiplying a price index and a volume index of the same type should equal a proportionate change in the current values.

## Understanding the Fisher Price Index

Fisher's Ideal Price Index is used to measure the price level and cost of living in the economy and is also used to calculate the inflation of an economy. The formula is the geometric mean of the Laspeyres and Paasch price indices. The formula of Fisher's Ideal Price Index is as follows:

Fisher Price Index = (Laspeyres Price Index * Paasche Price Index)^(0.5)

The index requires a decent amount of computations. In addition, the process is a little confusing, so it may be better to hear it written out:

1. First, you must calculate the Laspeyres Price Index for each period. You will need to remember that the Laspeyres Price Index uses observation prices and base quantities in the numerator and base prices and base quantities in the denominator.
2. The next step in the calculation is to determine the Paasche Price Index for each period. Like the Laspeyres Price Index, the Paasche Price Index uses observation prices and base quantities in the numerator and base prices and base quantities in the denominator.
3. The last step is to take the geometric mean of the Laspeyres and Paasche Price Indices in each period, resulting in the Fisher Price Index for that corresponding period.

Since the formula includes other indices, it's important to know how to calculate them too.

The formula for the Laspeyres price index is:

(𝚺 (Observation Price * Base Quantity)) / (𝚺 (Base Price * Base Quantity))

The formula for the Paasche Price Index is:

(𝚺 (Price at Observation Period * Observation Quantity)) / (𝚺 (Price at Base Period * Observation Quantity)) ## Fisher vs. Laspeyres vs. Paasche Price Index

The Laspeyres Price Index is a metric used to measure the change in the prices of a basket of goods and services relative to a specific base year/period weighting. It is an easy-to-calculate index and is pretty commonly used. It is also cheap to construct. An advantage of using the Laspeyres Price Index is that the quantities for any time in the future don't need to be calculated. Instead, only quantities from the base year are used.

The Laspeyres Price Index also presents a useful comparison because changes in the index result from price changes.

The main disadvantage of the index is that it has an upward bias, meaning that it tends to overstate price increases relative to other indices. As a result, the index tends to overestimate price levels and inflation.

This overestimation happens because of three different factors:

• An increase in newer goods and services which are more expensive causes an upward bias in price.
• The quality of the goods rises, increasing the cost of the goods. This price increase is not a result of inflation.
• Substitution of goods that are cheaper with goods that are considered to be more expensive.

The Paasche Price Index is used to measure the general price level, the cost of living in the economy and calculate inflation.

An advantage of using the Paasche Price Index is that it considers consumption patterns using current quantities. It is also not upward bias like the Laspeyres is with price increases.

The disadvantage of the Paasche Price Index is that it uses data on current quantities, which can be difficult to collect. It is also on the pricier side when compared to the Laspeyres Price Index.

Opposite the Laspeyres Price Index, the Paasche Price Index tends to underestimate the price changes. This is because the index already reflects changes in consumption patterns when consumers respond to changes in price.

Lastly, we have the Fisher Price Index, considered the best of the three indices.

Another name for the Fisher Price Index is the real index. This name perfectly describes the Fisher Price Index because of how it corrects for both biases in the two other indices.

Fisher's Ideal Price Index corrects for the upward bias of the Laspeyres Price Index and the downward bias of the Paasche Price Index.

It does this by taking the geometric average of both indices. For example, fisher's Ideal Price Index uses current and base year quantities as weights. Although the Fisher Price Index was created to eliminate the biases of the other indices, it still has some disadvantages. Fisher's Ideal Price Index is slightly more complex to construct than the Laspeyres and Paasche Price Index.

The quantities of the future years must be forecasted; with an index like Laspeyres, this isn't necessary. Therefore, only future prices need to be forecasted.

## Example

Below is an example of a hypothetical economy. In this example, the price and quantity will change depending on the year (year 0, year 1, year 2). Year 0 is designated as the base year.

Year 0:

As said above, Year 0 is the base year, so all the price indexes for that year will be 100. The calculations are shown below:

Year 0
Year 0Item XItem YItem Z
Price\$1.25\$2.50\$1.75
Quantity181227
Value\$19.25\$14.50\$28.75

Laspeyres Price Index = ((\$19.25 + \$14.50 + \$28.75) / (\$19.25 + \$14.50 + \$28.75)) * 100 = 100

Paasche Price Index = ((\$19.25 + \$14.50 + \$28.75) / (\$19.25 + \$14.50 + \$28.75)) * 100 = 100

Fisher Price Index = (100 * 100)^(0.5) = 100

Year 1:

Remember from above that Laspeyres Price Index uses observation prices and base year quantities in the numerator, and then in the denominator, it uses base year prices and quantities:

Year 1
Year 1Item XItem YItem Z
Price\$1.55\$3.40\$1.75
Quantity141730
Value\$15.55\$20.50\$31.75

Laspeyres Price Index = ((\$1.55*18)+(\$3.40*12)+(\$1.75*27)) / (\$19.25 + \$14.50 + \$28.75)) * 100 = 185.52

Remember that the Paasche Price Index uses observation price and quantities in the numerator and base year price and quantities in the denominator:

Paasche Price Index = ((\$15.55 + \$20.40 + \$31.75) / (\$19.25 + \$14.50 + \$28.75)) * 100 = 108.32

Fisher Price Index = (185.52 * 108.32)^(0.5) = 141.76

Year 2:

Remember from above that Laspeyres Price Index uses observation prices and base year quantities in the numerator, and then in the denominator, it uses base year prices and quantities:

Year 2
Year 2Item XItem YItem Z
Price\$1.75\$3.50\$1.90
Quantity191534
Value\$20.75\$18.50\$35.90

Laspeyres Price Index = ((\$1.75*18)+(\$3.50*12)+(\$1.90*27)) / (\$19.25 + \$14.50 + \$28.75)) * 100 = 199.68

Remember that the Paasche Price Index uses observation price and quantities in the numerator and base year price and quantities in the denominator:

Paasche Price Index = ((\$20.75 + \$18.50 + \$35.90) / (\$19.25 + \$14.50 + \$28.75)) * 100 = 120.24

Fisher Price Index = (199.68 * 120.24)^(0.5) = 154.95

Below is a summary of how the Laspeyres, Paasche, and Fisher Price Indices differ each year:

Summary
Price IndexYear 0Year 1Year 2
Laspeyres100.00185.52199.68
Paasche100.00108.32120.24
Fisher100.00141.76154.95

As we can see in the table and explained above, Fisher's Ideal Price Index falls in the middle of the Laspeyres and Paasche Price Indices. ### Everything You Need To Master Excel Modeling

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## What is the Consumer Price Index (CPI)?

The consumer price index measures the overall changes in consumer prices over time, based on a representative basket of goods and services bought by a typical consumer. CPI is calculated using a weighted average of prices for a basket of goods and services. All three indices mentioned above (Laspeyres, Paasche, and Fisher) are ways to measure the CPI of an economy. However, the most popular way that CPI is used is to gauge the inflation or deflation of an economy.

The Bureau of Labor Statistics (BLS) collects about 94,000 prices from over 20,000 retail and service providers to obtain the CPI. The index is said to cover 93% of the U.S. population.

The process of calculating the CPI is:

1. Fix the basket. The BLS surveys consumers to see what's in the typical consumer "shopping basket."

2. Find the prices. The BLS then collects data and information on the prices of all the goods and services in the basket.

3. Compute the basket's cost. Then the BLS or any individual can use the prices to compute the basket's total cost.

4. Choose a base year and compute the CPI. To determine the CPI, you use the formula below:

(Cost of Basket of Goods and Services (Current Year) / Cost of Basket (Base Year)) * 100

5. Compute the inflation rate: The formula to calculate the inflation rate is below:

(CPI this year - CPI last year) / CPI previous year)) * 100 ### Problems with using the CPI

While the government and others commonly use CPI to determine the inflation rate, it still has flaws. For example, over time, some items within the basket will rise faster than others.

Consumers will end up substituting expensive goods with relatively cheaper goods, mitigating the effects of price increases.

The CPI will miss the substitution by these consumers because it uses a fixed basket with fixed goods and services. This results in the CPI overstating increases in the cost of living. This phenomenon is known as the substitution bias.

Another problem arises with introducing new goods and services into the market. When new products are introduced into the market, there is more variety. This variety allows consumers to find products that more closely match their needs. This means, in a way, the dollar becomes more valuable. But, the CPI again misses this effect because it uses a fixed basket of goods and services. Meaning that, once again, the CPI overestimates increases in the cost of living.

One last problem is that the CPI doesn't accurately account for changes in the quality of goods and services. When products improve in quality, the value of a dollar increases too.

Even though the BLS tries to account for quality improvements, it cannot fully account for it because quality is hard to measure. So, once again, CPI overstates the increase in the cost of living.

As we can see, each of the problems with CPI makes it overstate the cost of living increases. The BLS has tried to make technical adjustments to fix this, but it's still estimated to overstate inflation by about 0.5 percent each year.

This is significant because CPI is used in many different ways that can affect the average consumer. For example, it is used in Social Security payments, and many various contracts have COLAs tied to the CPI. Therefore, an overstatement could be detrimental in these cases. ### Everything You Need To Master Financial Modeling

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Reviewed and edited by James Fazeli-Sinaki | LinkedIn