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Since inflation is impactful on the general price level of an economy, it is tantamount to measure inflation using a price index. As such, it is important to understand how a price index is modeled so that inflation rates derived from a price index can be precisely elucidated. This is because inflation is weighed as the percentage change in the price index.

## Price Index

The steps necessary for creating a price index include:

1. identification of the kind of index one wants to develop .i.e., yearly, monthly, or weekly;
2. identification of the basket of goods or services for both the previous and the current year, assuming one is interested in a yearly price index of a particular good;
3. calculation of the general price level of the previous and current year, respectively;
4. one then sets the price index for the base year to be 100. After that, one calculates the price index for the current year (the observation year); and
5. finally, to determine the inflation rate, one divides the price index for the current year by the price index for the base year minus 1.

A price index demonstrates the average prices of a collection of goods and services. Majority of the price indices around the globe use the Laspeyres method. It’s most common because the survey data on the consumption market is available with a lag. However, other methods such as the Paasche and Fisher methods are also used.

### Laspeyres Price Index

The Laspeyres price index measures the change in the price of the basket of goods relative to the base year weighting. It is given by:

$$I_{la}=\frac{∑P_t Q_0}{∑P_0 Q_0 }$$

Where

$$P_0$$= price of the individual item at the base year

$$Q_0$$= quantity of an individual item at the base period

$$P_t$$= price of the individual item at the observation period

### Paasche Price Index

The Paasche Price Index measures the change in the price and quantity of a basket of goods and services relative to base year price and observation year quantity. It is given by:

$$I_{pa}=\frac{∑P_t Q_t}{∑P_0 Q_t }$$

Where

$$Q_t$$=quantity of the individual item at the observation period

### Fisher’s Price Index

Fisher’s Price Index is the geometric mean of Laspeyres and Paasche price indices. That is:

$$I_{FI}=\sqrt{I_{la}\times I_{pa}}$$

### Example of Construction of Indices Used to Measure Inflation

Study the table below. It shows the prices and quantities of Slice bread and Rice in the years 2017 and 2018.

$$\begin{array}{c|c|c|c|c} \textbf{} & \textbf{Quantity (2017)} & \textbf{Price per Quantity (2017)} & \textbf{Quantity (2018)} & \textbf{Price per Quantity (2018)} \\ \hline \text{Sliced Bread} & 52 & \2.59 & 51 & \2.62 \\ \hline \text{Bag of Rice} & 36 & \0.89 & 38 & \0.85 \\ \end{array}$$

Calculate the Laspeyres, Paasche, and Fisher price indices and inflation rate taking 2017 as the base year.

Solution

The Laspeyres prices is given by:

$$I_{la}=\frac{∑P_t Q_0}{∑P_0 Q_0}=\frac{2.62 \times 52+0.85 \times 36}{52 \times 2.59+36 \times 0.89} \times 100=100.07$$

The inflation rate is the price index for the current year divided by the price index for the base year minus 1. Using the Laspeyres index, the inflation rate for the year 2018 is:

$$\frac{100.07}{100}-1=0.0007=0.07\%$$

The Paasche price index is given by:

$$I_{pa}=\frac{∑P_t Q_t}{∑P_0 Q_t }=\frac{2.62 \times 51+0.85 \times 38}{2.59 \times 51+0.89 \times 38} \times 100=100$$

Using the Paasche price index method, the inflation rate for the year 2018 is 0%.

The Fisher’s price index is the geometric mean of Laspeyres and Paasche indices, that is:

$$I_{FI}=\sqrt{I_{la}\times I_{pa}}=\sqrt{100.07×100}=100.03$$

Using the Fischer price index method, the inflation rate for the year 2018 is 3%.

### Question

The Laspeyres index gives an inflation rate of 12% for 2018. The Paasche price index method gives an inflation rate of 3%. What is the Fischer’s price index for 2018?

1. 3.87%
2. 6%
3. 7.5%

Solution

Fisher’s price index is the geometric mean of Laspeyres and Paasche price indices. That is:

$$I_{FI}=\sqrt{I_{la}\times I_{pa}} = \sqrt{I_{la}\times I_{pa}} = \sqrt{100.12\times 100.03} = 100.075$$

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