Construction of Index Numbers: Weighted Methods

Weighted Aggregate Methods: Introduction

A fundamental limitation of simple or unweighted index number methods (like the Simple Aggregate Method and the Simple Average of Relatives Method) is their failure to incorporate the relative importance of the different items included in the index. For instance, in a Consumer Price Index (CPI), a price change in essential goods like food grains or fuel should arguably have a greater impact on the overall index than a price change in less frequently purchased items like electronics or furniture, because essential goods constitute a larger portion of typical household expenditure.

Weighted index numbers are designed to overcome this deficiency by assigning specific weights to each commodity. These weights reflect the relative economic importance of the commodities within the group being measured (e.g., their share in total consumption expenditure, production value, or trade volume).

In weighted aggregate methods, the weights are typically based on the quantities (produced, consumed, or traded) of the commodities in either the base period or the current period, or some average of quantities.

The core principle of weighted aggregate price indices is to compare the total value (Price $\times$ Quantity) of a specific, consistently defined basket of goods across different periods. The quantity used as a weight for each commodity ($w_i$) determines its influence on the overall index.

Let's define the standard notation used in weighted aggregate methods:

• $p_{0i}$ = Price of the $i$-th commodity in the base period (denoted by 0)
• $q_{0i}$ = Quantity of the $i$-th commodity in the base period (denoted by 0)
• $p_{1i}$ = Price of the $i$-th commodity in the current period (denoted by 1)
• $q_{1i}$ = Quantity of the $i$-th commodity in the current period (denoted by 1)
• $w_i$ = Weight assigned to the $i$-th commodity (in weighted aggregate methods, this is usually a quantity: $q_{0i}$ or $q_{1i}$)
• The summation symbol ($\sum$) implies summing over all commodities $i$ included in the index.

Different specific methods arise based on the choice of quantity used as the weight ($w_i$). The two most prominent weighted aggregate price indices are the Laspeyres Index and the Paasche Index, which use base period quantities and current period quantities as weights, respectively.

Laspeyres Price Index ($P_L$)

Concept

The Laspeyres Price Index, named after the German economist Étienne Laspeyres, is a weighted aggregate price index that uses the quantities of the base period ($q_0$) as fixed weights for all periods. It measures the change in the cost of a basket of goods whose composition is held constant at the base period levels.

Essentially, it answers the hypothetical question: "If consumers (or producers) continued to buy (or produce) the exact same basket of goods as they did in the base period, how much would that basket cost in the current period compared to what it cost in the base period?"

Formula

The Laspeyres Price Index ($P_{01}^L$ or $L$) is calculated by finding the total value of the base period quantities at current period prices and dividing it by the total value of the base period quantities at base period prices, then multiplying by 100.

The formula is:

Let's break down the components:

• $\sum (p_{1i} q_{0i})$: This is the sum of the products of current period prices and base period quantities for all commodities. It represents the total cost of the base period basket valued at current period prices.
• $\sum (p_{0i} q_{0i})$: This is the sum of the products of base period prices and base period quantities for all commodities. It represents the total cost or total expenditure on the basket in the base period. This term is also the denominator and remains constant when calculating the index for different current periods against the same fixed base.

The summation ($\sum$) is carried out over all commodities $i=1, 2, \dots, N$.

Derivation Note: The formula is essentially a weighted average of price relatives, where the weight for each commodity is its base period value share ($p_{0i}q_{0i}$). While it's often presented as a direct aggregate ratio, it can be shown that $P_L = \sum \left( \frac{p_{1i}}{p_{0i}} \times 100 \right) \times W_i$, where $W_i = \frac{p_{0i}q_{0i}}{\sum p_{0i}q_{0i}}$.

Interpretation

Interpreting the Laspeyres Price Index value involves comparing it to the base index of 100:

• If the Laspeyres index is $130$, it means that the cost of purchasing the same fixed basket of goods that was consumed/bought in the base period is $130\%$ of its cost in the base period. This indicates a $30\%$ increase in the price level, as measured by the Laspeyres method.
• If the index is $95$, it means the cost is $95\%$ of the base period cost, indicating a $5\%$ decrease in price level.

Advantages

• Simplicity in Calculation (for subsequent periods): Once the base period quantities ($q_0$) and the base period total expenditure ($\sum p_{0i} q_{0i}$) are determined, calculating the Laspeyres index for subsequent periods only requires collecting current period prices ($p_1$). The weights ($q_0$) remain constant, making it relatively straightforward to compute the index series over time.
• Consistent Basket for Comparison: By holding the quantities fixed at base period levels, the Laspeyres index measures price change while keeping the "quantity component" of value change constant. This allows for a comparison of price changes for a consistent standard of living or production mix (as represented by the base period basket).
• Data Requirement: It only requires quantity data for the base period, which is often more readily available or less burdensome to collect than current period quantity data on an ongoing basis.

Disadvantages

• Substitution Bias (Upward Bias): This is the most significant limitation. The Laspeyres index uses fixed base period quantities. Over time, as prices of some goods rise relatively more than others, consumers tend to substitute towards relatively cheaper goods and away from expensive ones. The Laspeyres index does not account for this shift in consumption patterns. By continuing to use the larger base period quantities for goods that have become relatively expensive, it tends to overestimate the increase in the cost of living or the general price level. This is known as an "upward bias".
• Does Not Reflect Current Consumption Patterns: As time passes and tastes, technology, income levels, and availability of goods change, the base period basket ($q_0$) may become less representative of current consumption or production patterns. An index based on an outdated basket may not accurately reflect the price changes relevant to current economic behaviour.
• Requires Periodic Revision: Due to the outdated basket issue, the base period ($q_0$) and the composition of the basket must be periodically revised to maintain the relevance of the index.
• Issue with New Goods/Services: New commodities introduced after the base period cannot be easily incorporated into the fixed base period basket, making the index less comprehensive over time.

Despite its disadvantages, the Laspeyres formula is widely used, especially for indices like the Consumer Price Index (CPI) in many countries (though often with modifications or chaining to mitigate the bias and outdated basket issues), primarily due to its relative ease of calculation over time.

Answer:

Paasche Price Index ($P_P$)

Concept

The Paasche Price Index, named after the German economist Hermann Paasche, is a weighted aggregate price index that uses the quantities of the current period ($q_1$) as weights. It measures the change in the cost of purchasing a basket of goods whose composition is that of the current period, comparing its cost in the current period to what it would have cost in the base period.

It answers the hypothetical question: "How much does the basket of goods that is currently being bought (or produced) cost in the current period, compared to what it would have cost if it were bought (or produced) in the base period at base period prices?"

Formula

The Paasche Price Index ($P_{01}^P$ or $P$) is calculated by finding the total value of the current period quantities at current period prices and dividing it by the total value of the current period quantities hypothetically valued at base period prices, then multiplying by 100.

The formula is:

Let's break down the components:

• $\sum (p_{1i} q_{1i})$: This is the sum of the products of current period prices and current period quantities for all commodities. It represents the total cost or total expenditure on the basket in the current period. This is the numerator and changes with each current period for which the index is calculated.
• $\sum (p_{0i} q_{1i})$: This is the sum of the products of base period prices and current period quantities for all commodities. It represents the hypothetical cost of buying the current period's basket valued at base period prices. This term also changes with each current period, as $q_1$ varies.

The summation ($\sum$) is carried out over all commodities $i=1, 2, \dots, N$.

Derivation Note: Similar to Laspeyres, the Paasche formula can be seen as a weighted average of price relatives, but here the weight for each commodity is its current period value share ($p_{1i}q_{1i}$), scaled appropriately. Specifically, $P_P = \frac{100}{\sum W_i} \sum \left( \frac{p_{1i}}{p_{0i}} \times W_i \right)$ where $W_i = p_{0i}q_{1i}$.

Interpretation

Interpreting the Paasche Price Index value involves comparing it to the base index of 100:

• If the Paasche index is $130$, it means that the basket of goods actually consumed/bought in the current period cost $130\%$ of what that same basket would have cost if purchased in the base period. This indicates a $30\%$ increase in the price level relevant to the current period's consumption patterns.
• If the index is $95$, it means the cost is $95\%$ of the hypothetical base period cost, indicating a $5\%$ decrease in price level for the current basket.

Advantages

• Reflects Current Consumption Patterns: By using current period quantities ($q_1$) as weights, the Paasche index incorporates current consumption or production patterns and automatically accounts for shifts in quantities consumed (like substitution). This means it reflects the price change for the basket of goods that is currently relevant to consumers or producers.
• Accounts for Substitution: It implicitly reflects the substitution effect. If consumers have shifted towards relatively cheaper goods, the weights of these cheaper goods will be higher in the current period basket, which can lead the Paasche index to show a lower price increase than Laspeyres.

Disadvantages

• Data Collection Burden: A major practical difficulty is the requirement for collecting quantity data ($q_1$) for the current period for every commodity included in the index. This data needs to be collected for each period for which the index is calculated, which can be resource-intensive and costly, especially for indices covering a large number of items (like a national CPI).
• Lack of Comparability Over Time: Because the weights ($q_1$) change with every period, the basket of goods used in the calculation is different for each current period. This makes it difficult to make direct, period-to-period comparisons of price changes in the index, as the underlying basket of goods being priced is not consistent. The index measures the cost change of a *changing* basket.
• Substitution Bias (Downward Bias): While it accounts for substitution, it can exhibit a "downward bias" compared to Laspeyres. By assigning higher weights to items that consumers are buying more of in the current period (often because their relative prices have fallen or risen less), it can understate the price increase required to maintain the *base period* standard of living.
• Computationally More Complex (for time series): Calculating the Paasche index for a series of periods requires recalculating both the numerator and the denominator for each period, as $p_1$, $q_1$, and $\sum p_0 q_1$ all change.
• Issue with New Goods/Services: New commodities introduced after the base period can be included in the current period basket, but their prices would need to be estimated for the base period ($p_0$) to calculate $\sum p_0 q_1$, which can be challenging or require specific methodologies.

Despite its theoretical advantage of reflecting current consumption, the data requirements and the difficulty in making consistent period-to-period comparisons mean that the Paasche index is less frequently used as a primary reported index compared to Laspeyres or chained indices, although it is important for theoretical comparisons and in formulas like Fisher's Ideal Index.

Answer:

Fisher's Ideal Index (Price) ($P_F$)

Concept

The Fisher's Ideal Index, proposed by the American economist Irving Fisher, was developed to provide a compromise between the potentially upward-biased Laspeyres Index and the potentially downward-biased Paasche Index. It achieves this by taking the geometric mean of the Laspeyres and Paasche price indices.

Fisher called it "ideal" not because it's perfect, but because it satisfies certain desirable mathematical properties or tests that other indices often fail.

Formula

The Fisher's Ideal Price Index ($P_{01}^F$ or $F$) is calculated as the geometric mean of the Laspeyres Price Index ($P_{01}^L$) and the Paasche Price Index ($P_{01}^P$).

Alternatively, substituting the full formulas for Laspeyres and Paasche (without the $\times 100$ part for the internal calculation, then multiplying by 100 at the end):

where $\frac{\sum p_1 q_0}{\sum p_0 q_0}$ is the Laspeyres price ratio and $\frac{\sum p_1 q_1}{\sum p_0 q_1}$ is the Paasche price ratio (without the $\times 100$). The final multiplication by 100 converts the ratio back into a percentage index.

Why is it Called "Ideal"? (Properties Satisfied)

Fisher's index is considered "ideal" because it satisfies two important tests of consistency for index numbers:

• Time Reversal Test: This test requires that if an index is calculated from period 0 to period 1 ($I_{01}$), the index calculated from period 1 to period 0 ($I_{10}$) should be the reciprocal of the first index. Mathematically, $I_{01} \times I_{10} = 1$ (if calculated as a ratio) or $I_{01} \times I_{10} = 10000$ (if calculated as percentage indices). Laspeyres Index ($P_{01}^L = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$) and its reverse ($P_{10}^L = \frac{\sum p_0 q_1}{\sum p_1 q_1} \times 100$) do not satisfy this (i.e., $P_{01}^L \times P_{10}^L \neq 10000$). Similarly for Paasche ($P_{01}^P = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$) and its reverse ($P_{10}^P = \frac{\sum p_0 q_0}{\sum p_1 q_0} \times 100$), $P_{01}^P \times P_{10}^P \neq 10000$. However, Fisher's Ideal Index satisfies the Time Reversal Test: $P_{01}^F \times P_{10}^F = 10000$.
• Factor Reversal Test: This test requires that the product of a price index and the corresponding quantity index (both calculated using the same method) should be equal to the corresponding value index ($V_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_0} \times 100$). The Fisher's Ideal Quantity Index is $Q_{01}^F = \sqrt{\left( \frac{\sum q_1 p_0}{\sum q_0 p_0} \right) \times \left( \frac{\sum q_1 p_1}{\sum q_0 p_1} \right)} \times 100$. Fisher's Price Index and Fisher's Quantity Index satisfy the Factor Reversal Test: $\frac{P_{01}^F \times Q_{01}^F}{100} = V_{01}$. Laspeyres ($P_{01}^L \times Q_{01}^L / 100 \neq V_{01}$) and Paasche ($P_{01}^P \times Q_{01}^P / 100 \neq V_{01}$) indices do not satisfy this test.

By satisfying these tests, Fisher's index is considered mathematically superior and conceptually sounder for capturing overall change, as it treats price and quantity changes symmetrically.

Disadvantages

• Computationally Complex: Calculating Fisher's index is more demanding than calculating either Laspeyres or Paasche individually, as it requires calculating both indices first and then finding their geometric mean. This makes it less practical for real-time calculation for large numbers of items.
• High Data Requirement: It requires quantity data for both the base period ($q_0$) and the current period ($q_1$) for all included commodities, which is often difficult and costly to collect regularly.
• Interpretation is Less Intuitive: The interpretation of a geometric mean is less straightforward than the simple aggregate cost comparison of Laspeyres (fixed base basket) or Paasche (current basket). It doesn't directly represent the change in cost of a clearly defined physical basket of goods in the same way as the other two indices.

Due to its computational and data requirements, Fisher's Ideal Index is less commonly used as a primary index for frequent publication (like monthly CPI). However, it is often used for research, analytical purposes, or as a benchmark for evaluating other index number methods. In some cases, chained versions of Laspeyres or Paasche indices are used in practice, which approximate the properties of Fisher's index.

Answer:

Marshall-Edgeworth Index (Price) ($P_{ME}$)

Concept

The Marshall-Edgeworth Index, named after economists Alfred Marshall and F.Y. Edgeworth, is another type of weighted aggregate price index. It aims to address the limitations of using only base period quantities (Laspeyres) or only current period quantities (Paasche) as weights. It does so by employing an average (specifically, the sum) of the quantities from both the base period and the current period as the weights.

Formula

The Marshall-Edgeworth Price Index ($P_{01}^{ME}$) is calculated by taking the ratio of the total value of a basket consisting of the sum of base and current period quantities, first valued at current period prices, and then valued at base period prices. This ratio is then multiplied by 100.

The formula is:

Where:

• $p_{1i}$ = Price of the $i$-th commodity in the current period.
• $p_{0i}$ = Price of the $i$-th commodity in the base period.
• $q_{1i}$ = Quantity of the $i$-th commodity in the current period.
• $q_{0i}$ = Quantity of the $i$-th commodity in the base period.
• The term $(q_{0i} + q_{1i})$ serves as the weight for the $i$-th commodity, representing the sum of quantities in the base and current periods.
• The summation ($\sum$) is over all commodities $i$.

The formula can be expanded as:

Equation (ii) shows that the Marshall-Edgeworth index's numerator is the sum of the Laspeyres numerator ($\sum p_1 q_0$) and the Paasche numerator ($\sum p_1 q_1$, which is also $\sum p_1 q_1$). The denominator is the sum of the Laspeyres denominator ($\sum p_0 q_0$) and the Paasche denominator ($\sum p_0 q_1$). This structure highlights its nature as a compromise between Laspeyres and Paasche.

Properties and Comparison

• Weighting Scheme: It uses an average of base and current period quantities ($(q_0 + q_1)/2$ - although the $/2$ cancels out in the ratio, the weighting is proportional to $q_0+q_1$) as weights. This provides a weighting that is potentially more representative of consumption/production patterns over the two periods being compared than using quantities from just one period.
• Data Requirement: Like the Paasche and Fisher indices, it requires quantity data for both the base period ($q_0$) and the current period ($q_1$) for all commodities, which can be a significant data collection burden.
• Bias: The Marshall-Edgeworth index is generally considered to lie between the Laspeyres and Paasche indices, potentially reducing the upward bias of Laspeyres and the downward bias of Paasche. However, it does not possess the strong theoretical properties of Fisher's Ideal Index.
• Tests of Consistency: The Marshall-Edgeworth index satisfies the Time Reversal Test ($P_{01}^{ME} \times P_{10}^{ME} = 10000$). However, it does not satisfy the Factor Reversal Test ($P_{01}^{ME} \times Q_{01}^{ME} / 100 \neq V_{01}$).
• Usage: While theoretically interesting and satisfying the Time Reversal Test, the Marshall-Edgeworth index is not as widely used in practice for major official statistics compared to Laspeyres (or modified Laspeyres) and sometimes Paasche, and less used in research than Fisher's Ideal Index.

Answer:

Weighted Average of Price Relatives Method

This method is an alternative way to construct a weighted price index. Instead of aggregating the total costs of baskets (as in Laspeyres, Paasche, etc.), it works with the individual price relatives of each commodity. Each commodity's price relative is weighted according to the commodity's importance, and then a weighted average (usually the arithmetic mean) of these weighted relatives is calculated.

Concept

Recall that a simple average of price relatives gives equal weight to the relative price change of every commodity. The Weighted Average of Price Relatives method refines this by assigning varying weights to each commodity's price relative ($P_i = \frac{p_{1i}}{p_{0i}} \times 100$). The weights ($w_i$) are chosen to reflect the economic significance of the $i$-th commodity relative to the total group of commodities.

Choice of Weights ($w$)

The weights ($w_i$) in this method are typically based on the value of the commodity in the base period or sometimes the current period. The most common and theoretically significant choice is using the base period value of the commodity as the weight:

$$w_i = p_{0i} q_{0i}$$

This represents the total expenditure (or revenue) on commodity $i$ in the base period. Using base period value weights gives more importance to commodities that had a larger share in the total expenditure (or value of production/trade) in the base period.

Other choices for weights could include current period value ($p_{1i} q_{1i}$), or an average of base and current values or quantities, but base period value weights are the standard for the specific form of this method that relates to other index numbers.

Formula (Using Weighted Arithmetic Mean)

The Weighted Average of Price Relatives Index ($P_{01}$) using the weighted arithmetic mean is calculated as the sum of the products of each price relative and its corresponding weight, divided by the sum of the weights.

where:

• $P_i = \frac{p_{1i}}{p_{0i}} \times 100$ is the price relative for the $i$-th commodity.
• $w_i$ is the weight assigned to the $i$-th commodity (typically $p_{0i} q_{0i}$).
• $\sum (P_i w_i)$ is the sum of the weighted price relatives over all commodities $i=1, \dots, N$.
• $\sum w_i$ is the sum of all the weights, which is the total value of the basket in the base period if $w_i = p_{0i} q_{0i}$.

Relationship to Laspeyres Index

A crucial point to note is that when the weights used in the Weighted Average of Price Relatives Method (using Arithmetic Mean) are the base period values ($w_i = p_{0i} q_{0i}$), the resulting index number is mathematically identical to the Laspeyres Price Index.

Proof of Equivalence:

Start with the formula for the Weighted Average of Price Relatives using base period value weights:

Substitute the formula for the price relative $P_i = \frac{p_{1i}}{p_{0i}} \times 100$ and the weight $w_i = p_{0i} q_{0i}$:

In the numerator, the $p_{0i}$ terms cancel out for each commodity $i$:

This simplifies the numerator:

Since 100 is a constant, it can be taken outside the summation:

This final expression is the exact formula for the Laspeyres Price Index ($P_{01}^L$).

Thus, calculating the Weighted Average of Price Relatives using base period value weights yields the same result as the Laspeyres Price Index.

Weighted Geometric Mean

Similar to the simple average of relatives, a weighted geometric mean of price relatives can also be calculated. This involves taking the logarithm of each price relative, multiplying by its weight, summing these products, dividing by the sum of weights, and finally taking the antilog of the result.

Where $P_i = \frac{p_{1i}}{p_{0i}} \times 100$ and $w_i$ are the chosen weights. If base period value weights ($w_i = p_{0i}q_{0i}$) are used with the Weighted Geometric Mean of Relatives, the result is related to Fisher's Ideal Index but is not identical to it. However, using specific weights ($w_i = \frac{p_{0i}q_{0i}}{\sum p_{0i}q_{0i}}$, the base period value shares) with the Weighted Geometric Mean of Price Relatives results in the Törnqvist Index, which is another index often considered better than Laspeyres or Paasche and used in some official statistics.

Answer: