Construction of Index Numbers: Simple Methods

Simple Aggregate Method (Price and Quantity)

This is one of the most basic methods for constructing an index number, particularly for prices. It involves comparing the total value of a fixed set of observations (like prices or quantities) for a group of commodities in the current period with the total value of the same observations in the base period. It's called 'simple' because it does not involve any explicit weighting based on the importance of different commodities.

Simple Aggregate Price Index

The simple aggregate price index measures the relative change in the sum of the prices of a specified group of commodities between the base period and the current period. It is calculated by taking the ratio of the sum of current period prices to the sum of base period prices, and multiplying by 100.

• Formula: The formula for the Simple Aggregate Price Index ($P_{01}$) is:

  $$P_{01} = \frac{\sum p_1}{\sum p_0} \times 100$$

  ... (i)

  where:
  • $P_{01}$ = Simple Aggregate Price Index for the current period (denoted by 1) based on the base period (denoted by 0).
  • $\sum p_1$ = Sum of the prices of all included commodities in the group during the current period. If there are $n$ commodities, this is $\sum_{i=1}^{n} p_{1i}$, where $p_{1i}$ is the price of the $i$-th commodity in the current period.
  • $\sum p_0$ = Sum of the prices of all included commodities in the group during the base period. If there are $n$ commodities, this is $\sum_{i=1}^{n} p_{0i}$, where $p_{0i}$ is the price of the $i$-th commodity in the base period.
• Calculation Procedure: To calculate this index, you need to:
  1. Collect the prices of all selected commodities for the base period and sum them up ($\sum p_0$).
  2. Collect the prices of the same set of commodities for the current period and sum them up ($\sum p_1$).
  3. Apply the formula: divide the sum of current prices by the sum of base prices and multiply by 100.

Answer:

Simple Aggregate Quantity Index

Similarly, a simple aggregate quantity index measures the relative change in the sum of the quantities of a group of commodities between the base period and the current period. This is calculated as the ratio of the sum of current period quantities to the sum of base period quantities, multiplied by 100.

• Formula: The formula for the Simple Aggregate Quantity Index ($Q_{01}$) is:

  $$Q_{01} = \frac{\sum q_1}{\sum q_0} \times 100$$

  ... (ii)

  where:
  • $Q_{01}$ = Simple Aggregate Quantity Index for the current period (1) based on the base period (0).
  • $\sum q_1$ = Sum of the quantities of all included commodities in the group during the current period ($\sum_{i=1}^{n} q_{1i}$).
  • $\sum q_0$ = Sum of the quantities of all included commodities in the group during the base period ($\sum_{i=1}^{n} q_{0i}$).
• Important Note: This method for quantity index is only statistically and practically meaningful if all quantities are expressed in the same unit or comparable units. For example, summing quantities of different agricultural products all measured in 'tonnes' or 'quintals' might be acceptable, but directly adding quantities like 'kg' (for wheat) and 'litres' (for milk) and 'meters' (for cloth) together as $\sum q_0$ or $\sum q_1$ does not represent a physically meaningful aggregate quantity and thus makes the resulting index value difficult to interpret correctly.

Limitations of Simple Aggregate Method

Despite its simplicity, the Simple Aggregate Method suffers from significant drawbacks that limit its practical application in constructing meaningful index numbers, especially for diverse groups of commodities:

• Dependency on Units: The index value is heavily influenced by the units in which prices (or quantities) are quoted. If the unit of measurement for even one commodity is changed (e.g., from price per kg to price per gram, or from quantity in tonnes to quantity in kilograms), the sum will change drastically, and that commodity's relative influence on the index will be disproportionately altered, even if its price per unit or quantity remains constant. This unit dependency makes comparisons over time or across different datasets unreliable if units are not uniform.
• Implicit Weighting Bias: This method implicitly assigns weights based on the absolute prices (or quantities) of the commodities, not their economic importance. An item with a higher price (e.g., a luxury car) will have a much larger impact on the sum and thus on the index than an item with a low price (e.g., a loaf of bread), regardless of how much of each item is bought or sold in the economy. This can lead to a distorted picture of the overall price or quantity change, as items that are economically less significant but have high prices can dominate the index movement.
• Not Suitable for Diverse Quantities: As highlighted earlier, the Simple Aggregate Quantity Index is generally not appropriate when the commodities involved are measured in fundamentally different physical units (like weight, volume, length). Summing such diverse quantities does not yield a meaningful total.
• Does Not Account for Relative Changes: The method considers only the absolute sum of prices/quantities. It does not take into account the relative price/quantity changes of individual items. For instance, a $10\%$ increase in the price of an expensive item might contribute more to the sum than a $50\%$ increase in the price of a cheaper item, even if the $50\%$ change is more significant in terms of relative impact.

Due to these severe limitations, especially the unit dependency and implicit arbitrary weighting, the Simple Aggregate Method is rarely used for constructing official or widely-used index numbers like CPI, WPI, or IIP. Weighted aggregate methods (like Laspeyres, Paasche, Fisher), which account for the economic importance of commodities, are preferred in practice.

Simple Average of Price Relatives Method

This method offers an alternative to the simple aggregate method by first converting all prices into relatives (percentages relative to the base period). It then calculates an average (typically the arithmetic mean or geometric mean) of these individual price relatives to arrive at an overall index number. This approach addresses the unit dependency issue inherent in the simple aggregate method.

Method using Arithmetic Mean (AM)

The most common way to calculate the simple average of price relatives index is by using the arithmetic mean. The steps involved are:

1. For each commodity in the group, calculate its Price Relative ($P$) by comparing its price in the current period ($p_1$) to its price in the base period ($p_0$) and multiplying by 100: $P = \frac{p_1}{p_0} \times 100$.
2. Sum up all the calculated price relatives for all commodities ($\sum P$).
3. Divide the sum of price relatives by the total number of commodities ($N$) included in the index to get the simple arithmetic average.

• Formula: The formula for the Simple Average of Price Relatives Index using Arithmetic Mean ($P_{01}$) is:

  $$P_{01} = \frac{\sum P}{N} = \frac{1}{N} \sum_{i=1}^{N} \left( \frac{p_{1i}}{p_{0i}} \times 100 \right)$$

  ... (i)

  where:
  • $P_{01}$ = Simple Average of Price Relatives Index for the current period (1) based on the base period (0).
  • $P_i = \frac{p_{1i}}{p_{0i}} \times 100$ = Price relative for the $i$-th commodity.
  • $\sum P$ = Sum of the price relatives for all commodities ($\sum_{i=1}^{N} P_i$).
  • $N$ = Total number of commodities included in the index.
  • $p_{1i}$ = Price of the $i$-th commodity in the current period.
  • $p_{0i}$ = Price of the $i$-th commodity in the base period.
• Advantage: A key advantage of this method is that the resulting index value is independent of the units in which the prices of individual commodities are quoted. Since the price relative is a ratio ($p_1/p_0$), the units cancel out, making the relative a pure number (percentage). Averaging these pure numbers removes the unit bias seen in the simple aggregate method.

Answer:

Method using Geometric Mean (GM)

While the arithmetic mean is easier to calculate, the geometric mean is sometimes considered more appropriate for averaging ratios and percentages like price relatives. This is because the geometric mean provides consistent results regardless of which period is taken as the base year and is less affected by extreme values compared to the arithmetic mean.

• Formula: The formula for the Simple Average of Price Relatives Index using Geometric Mean ($P_{01}$) is the $N$-th root of the product of the $N$ price relatives:

  $$P_{01} = \left( P_1 \times P_2 \times \dots \times P_N \right)^{1/N} = \left( \prod_{i=1}^{N} P_i \right)^{1/N}$$

  ... (ii)

  Or, using logarithms for easier calculation (especially with many commodities):

  $$P_{01} = \text{Antilog} \left( \frac{1}{N} \sum_{i=1}^{N} \log P_i \right)$$

  ... (iii)

  where $\prod_{i=1}^{N} P_i$ denotes the product of the price relatives for all $N$ commodities.
• Advantage: The geometric mean satisfies the Time Reversal Test (meaning if you calculate the index from period 0 to 1 and then from 1 to 0, the results will be reciprocal), which the arithmetic mean does not. It is also less influenced by outliers.
• Disadvantage: Calculation using the geometric mean is generally more complex, especially without a calculator or logarithmic tables/software.

(Calculating the index for Example 1 data using Geometric Mean would involve taking the 4th root of the product of the four price relatives: $(133.33 \times 122.22 \times 112.50 \times 120.00)^{1/4}$. This would yield a value slightly lower than the arithmetic mean result of 122.01).

Limitations of Simple Average of Relatives Method

While this method improves upon the simple aggregate method by being unit-free, it still has notable limitations:

• Lack of Weighting: The most significant drawback is that it gives equal importance (weight) to the price relative of each commodity, regardless of its economic significance (e.g., how much is consumed or produced). A $10\%$ price increase for a commodity like salt, which constitutes a tiny part of a household budget, has the same impact on the index as a $10\%$ price increase for rice, a major staple. This lack of weighting makes the index less representative of overall price changes from an economic perspective.
• Arithmetic Mean Bias: When using the Arithmetic Mean, items experiencing very large percentage price increases (having high price relatives) can disproportionately inflate the average, potentially overstating the overall price change. The Geometric Mean mitigates this to some extent but adds computational complexity.
• Cannot be Used for Quantity Index: The method is primarily discussed in the context of price indices. While one could calculate quantity relatives and average them, a simple average of quantity relatives suffers from similar weighting issues and is not commonly used. Weighted quantity indices are standard.

Due to the crucial limitation of lacking a proper weighting system based on economic importance, the Simple Average of Relatives Method is, like the Simple Aggregate Method, considered an unweighted index method and is less frequently used in practice for constructing major economic indices compared to weighted methods.

Limitations of Simple Index Number Methods

Both the Simple Aggregate Method (I1) and the Simple Average of Price Relatives Method (I2) belong to the category of 'simple' or 'unweighted' index number methods. While they are easy to understand and compute, they suffer from significant drawbacks that render them unsuitable for constructing reliable and representative index numbers for complex economic phenomena involving multiple commodities.

The key limitations common to or specific to these simple methods are:

1. Influence of Units (Specific to Simple Aggregate Method): The Simple Aggregate Index is directly and heavily influenced by the units in which the prices or quantities of commodities are expressed. Changing the unit of measurement for even a single item (e.g., quoting price per quintal instead of per kg for wheat) can drastically change the sum and thus the index value, even if the intrinsic value or price relationship remains the same. This makes the index value arbitrary and dependent on how the data is presented, severely limiting its comparability and reliability unless units are perfectly standardized and uniform, which is often not practical across diverse commodities.
2. Influence of Absolute Values / Implicit Weighting (Specific to Simple Aggregate Method): In the Simple Aggregate Method, commodities with higher absolute prices (for price index) or quantities (for quantity index) have a disproportionately larger weight or influence on the total sum and, consequently, on the final index value. A commodity with a high price, regardless of how little it is consumed or traded, will dominate the index movement compared to a low-priced but widely consumed commodity. This 'weighting' is implicit and based purely on absolute values, not on the economic importance or volume of transactions.
3. Lack of Economic Weighting (Common to Both Methods): This is the most fundamental limitation. Neither the Simple Aggregate Method nor the Simple Average of Relatives Method takes into account the actual economic importance or relative significance of the different commodities included in the index. For a price index (like CPI), a commodity that constitutes a large portion of household expenditure (like rice or fuel) should logically have a greater impact on the index than a commodity consumed rarely or in small quantities (like a specific spice or luxury item). Simple methods fail to reflect this reality. They either weight by absolute price (Simple Aggregate) or give equal weight to each commodity's percentage change (Simple Average of Relatives), neither of which represents true economic importance.
4. Arithmetic Mean Bias (Specific to Simple Average of Relatives Method using AM): When the arithmetic mean is used to average price relatives, the index tends to be upward biased. Items that experience very large percentage price increases (often outliers or less important items) can significantly pull up the average, potentially overstating the overall increase in the price level. The Geometric Mean mitigates this bias, but its calculation is more complex.
5. Aggregation Issues for Quantities (Specific to Simple Aggregate Method): As discussed earlier, applying the Simple Aggregate Method to quantities is generally illogical and yields a meaningless result unless all commodities are measured in identical physical units. Summing quantities of items measured in kilograms, litres, meters, and numbers together does not represent a coherent total quantity.

Because of these significant weaknesses, particularly the failure to incorporate the relative economic importance (weight) of different items, simple index number methods are considered elementary and are generally not used for constructing important official index numbers. They serve primarily as an introduction to the concept of index numbers. More sophisticated weighted index numbers (such as Laspeyres' Index, Paasche's Index, Fisher's Ideal Index, and Marshall-Edgeworth Index) were developed precisely to overcome the limitations of simple methods by explicitly assigning weights based on the economic significance (e.g., consumption or production quantities) of each commodity.