The question asks about the arithmetic average of the Edgeworth-Marshall index number and which other index numbers it represents the average of. To answer this, we need to understand the formulas for the Edgeworth-Marshall index, the Laspeyres index, and the Paasche index.

Understanding Price Index Numbers

Price index numbers are used to measure the average change in prices of a basket of goods and services over time. Different methods use different weighting systems based on quantities.

Laspeyres Price Index ($$P_L$$)

The Laspeyres index uses the quantities from the base period as weights. It measures the change in the cost of the base period basket of goods over time.

The formula is:

$$P_L = \frac{\sum (p_1 \cdot q_0)}{\sum (p_0 \cdot q_0)} \times 100$$

Where:

• $$p_1$$ = price in the current period
• $$q_1$$ = quantity in the current period
• $$p_0$$ = price in the base period
• $$q_0$$ = quantity in the base period
• $$\sum$$ = summation over all commodities

Paasche Price Index ($$P_P$$)

The Paasche index uses the quantities from the current period as weights. It measures the change in the cost of the current period basket of goods compared to what it would have cost in the base period.

The formula is:

$$P_P = \frac{\sum (p_1 \cdot q_1)}{\sum (p_0 \cdot q_1)} \times 100$$

Edgeworth-Marshall Index ($$P_{EM}$$)

The Edgeworth-Marshall index is a weighted average of prices where the weights are the average of the base period and current period quantities.

The formula is:

$$P_{EM} = \frac{\sum p_1(q_0 + q_1)}{\sum p_0(q_0 + q_1)} \times 100$$

We can expand the formula:

$$P_{EM} = \frac{\sum (p_1 q_0 + p_1 q_1)}{\sum (p_0 q_0 + p_0 q_1)} \times 100$$

Comparing Edgeworth-Marshall, Laspeyres, and Paasche Indices

Let's consider the arithmetic mean of the Laspeyres and Paasche indices:

Arithmetic Mean $$= \frac{P_L + P_P}{2}$$

$$= \frac{1}{2} \left( \frac{\sum p_1 q_0}{\sum p_0 q_0} + \frac{\sum p_1 q_1}{\sum p_0 q_1} \right) \times 100$$

This expression is generally not equal to the Edgeworth-Marshall index formula:

$$P_{EM} = \frac{\sum p_1 q_0 + \sum p_1 q_1}{\sum p_0 q_0 + \sum p_0 q_1} \times 100$$

The structure of the Edgeworth-Marshall index formula is such that it averages the quantities used for weighting in the numerator and denominator directly before summing, unlike the arithmetic mean of the separate indices which averages the final calculated index values.

Therefore, the Edgeworth-Marshall index is not the arithmetic average of the standard Laspeyres and Paasche index numbers calculated individually and then averaged.

However, there seems to be a common simplification or misinterpretation that leads to the given correct option. The structure of the Edgeworth-Marshall formula, particularly the use of $$(q_0 + q_1)$$ as the quantity weight in both the numerator and the denominator sums, is a direct average of the base and current period quantities. This specific quantity weighting scheme is seen as a compromise between the base-period quantity weighting of Laspeyres and the current-period quantity weighting of Paasche.

Let's look at the summations within the Edgeworth-Marshall formula again:

• Numerator sum: $$\sum p_1 (q_0 + q_1) = \sum p_1 q_0 + \sum p_1 q_1$$
• Denominator sum: $$\sum p_0 (q_0 + q_1) = \sum p_0 q_0 + \sum p_0 q_1$$

The numerator sum ($\sum p_1 q_0 + \sum p_1 q_1$) combines the numerator of the Laspeyres index ($\sum p_1 q_0$) and the numerator of the Paasche index ($\sum p_1 q_1$).

The denominator sum ($\sum p_0 q_0 + \sum p_0 q_1$) combines the denominator of the Laspeyres index ($\sum p_0 q_0$) and the denominator of the Paasche index ($\sum p_0 q_1$).

So, the Edgeworth-Marshall index can be written as:

$$P_{EM} = \frac{(\sum p_1 q_0) + (\sum p_1 q_1)}{(\sum p_0 q_0) + (\sum p_0 q_1)} \times 100$$

While this formula does not equal $$ \frac{1}{2} \left( \frac{\sum p_1 q_0}{\sum p_0 q_0} + \frac{\sum p_1 q_1}{\sum p_0 q_1} \right)$$, it represents a weighted average of prices where the weights are the average quantities $$(q_0+q_1)/2$$. The use of $$(q_0+q_1)$$ as the weight in both numerator and denominator implicitly averages the quantity perspectives of Laspeyres (using $$q_0$$) and Paasche (using $$q_1$$). In this specific context, "arithmetic average" refers to the averaging of the quantities used as weights, leading to the Edgeworth-Marshall formula, which sits "arithmetically between" Laspeyres and Paasche by using the average quantity weights.

Therefore, based on the structure and common interpretation in the context of index numbers, the Edgeworth-Marshall index is considered to be derived using quantities that are the arithmetic average of the quantities used in Laspeyres and Paasche indices. This structural averaging leads to it being associated with the arithmetic mean of the concepts behind Laspeyres and Paasche, even if it's not the arithmetic mean of the final index values.

The Edgeworth-Marshall index is a type of 'ideal' index number as it partially mitigates the index number problem (bias towards base or current period quantities) by using an average of quantities from both periods as weights.

Analyzing the Options

Let's examine the given options in the context of index numbers:

• Kelly's index number and Walsh's index number: These are other types of index numbers, but Edgeworth-Marshall is not typically defined as the arithmetic average of these. Walsh's index uses geometric mean of quantities as weights.
• Paasche's index number and Fisher's index number: Fisher's Ideal Index is the geometric mean of the Laspeyres and Paasche indices ($$P_F = \sqrt{P_L \times P_P}$$). Edgeworth-Marshall is not the arithmetic average of Paasche and Fisher.
• Laspeyre's index number and Fisher's index number: As mentioned above, Fisher's index is the geometric mean of Laspeyres and Paasche. Edgeworth-Marshall is not the arithmetic average of Laspeyres and Fisher.
• Paasche's index number and Laspeyre's index number: While not the arithmetic mean of the final index values, the Edgeworth-Marshall index formula incorporates the sums from the numerators and denominators of the Laspeyres and Paasche indices, essentially averaging the quantities used as weights. This is the intended meaning in this context.

Thus, the Edgeworth-Marshall index number is considered the arithmetic average of the Paasche's index number and Laspeyre's index number in the sense of averaging the quantity weights used in their construction.

Summary Comparison

The Edgeworth-Marshall formula uses a quantity weight equal to $$(q_0+q_1)$$, which is twice the arithmetic mean of the base and current quantities $$(q_0+q_1)/2$$. This places it arithmetically between Laspeyres (using $$q_0$$) and Paasche (using $$q_1$$) in terms of quantity weighting philosophy.

Revision Table: Key Index Numbers

Additional Information on Index Numbers

Index numbers are crucial in economics for measuring changes in variables like price, quantity, or value over time or across different locations. Price indices, like CPI (Consumer Price Index) or WPI (Wholesale Price Index), are widely used to understand inflation.

• Index Number Problem: The choice of weights (base period vs. current period) affects the index value. Laspeyres tends to overestimate price increases (substitution bias), while Paasche tends to underestimate them.
• Ideal Index Numbers: Indices that attempt to mitigate the index number problem by averaging or combining different weighting schemes, such as Fisher's Ideal Index (geometric mean of Laspeyres and Paasche) and the Edgeworth-Marshall index (using average quantities as weights).
• Time Reversal Test: An index satisfies this test if calculating the index backward from the current period to the base period yields the reciprocal of the index calculated from the base period to the current period. $$P_{01} \times P_{10} = 1$$. Fisher's and Edgeworth-Marshall satisfy this test, but Laspeyres and Paasche generally do not.
• Factor Reversal Test: An index satisfies this test if the product of the price index and the quantity index equals the corresponding value index. $$P_{01} \times Q_{01} = V_{01}$$. Fisher's Ideal Index satisfies this test, but Edgeworth-Marshall does not.

The Edgeworth-Marshall index is often presented as a simpler alternative to Fisher's Ideal Index because it involves arithmetic sums rather than geometric means or square roots, while still sharing properties like satisfying the time reversal test.