Index Numbers and Time Based Data

i) Correct. An index number is a ratio (or percentage) of two values, so all units cancel out, making it a pure (dimensionless) number.

ii) Correct. Because index numbers are ratios, they do not depend on the units in which prices or quantities are measured.

iii) Incorrect. An index number is always a ratio of two positive quantities multiplied by 100, so it is always a non-negative number.

iv) Correct. As the wholesale price index rises, the same amount of money buys fewer goods, meaning the purchasing power of money falls.

Correct Answer: iii) Weighted aggregative price index

A weighted aggregative price index uses the prices of items weighted by their quantities (or relative importance), giving a composite measure of price change. This distinguishes it from a simple aggregative index (no weights) or price relatives (individual ratios).

Correct Answer: iv) Paasche's index

Paasche's price index uses current-period quantities as weights:
$$P_{\text{Paasche}} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$$
where $q_1$ denotes current-year quantities. Laspeyres, by contrast, uses base-period quantities.

Correct Answer: i) Quantity index

A quantity index measures the change in the volume/quantity of goods produced, consumed, or traded over time, keeping prices constant. It is defined as:
$$Q_{01} = \frac{\sum q_1 p_0}{\sum q_0 p_0} \times 100$$

Correct Answer: iii) Laspeyres method

Laspeyres price index uses base-period quantities ($q_0$) as weights:
$$P_{\text{Laspeyres}} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$$

*(Note: The answer key in the textbook states i), but by definition Laspeyres uses base-period quantities. Fisher's method is the geometric mean of Laspeyres and Paasche's. The standard correct answer is iii) Laspeyres method.)*

Correct Answer: iv) 100

By convention, the index number of the base period is always taken as 100, since the price (or quantity) is compared with itself:
$$I_0 = \frac{p_0}{p_0} \times 100 = 100$$

Correct Answer: i) Average

An index number is a specialised average (typically a weighted average of price relatives or aggregates) that summarises the overall change in a group of related variables over time or space.

Correct Answer: i) Percentage

Index numbers are conventionally expressed as percentages. The formula multiplies the ratio by 100:
$$I = \frac{p_1}{p_0} \times 100$$
so the result is always a percentage figure (e.g., 120 means a 20% increase over the base).

Correct Answer: iii) Fisher

Fisher's index is called the Ideal Index because:
1. It is the geometric mean of Laspeyres and Paasche's indices, thus balancing the upward bias of Laspeyres and the downward bias of Paasche's.
2. It satisfies both the Time Reversal Test and the Factor Reversal Test.
$$P_{\text{Fisher}} = \sqrt{P_{\text{Laspeyres}} \times P_{\text{Paasche}}}$$

Correct Answer: i) Quantity in base year

Laspeyres price index:
$$P_L = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$$
The weight used is $q_0$, i.e., the quantity in the base year.

Correct Answer: ii) Quantity in current year

Paasche's price index:
$$P_P = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$$
The weight used is $q_1$, i.e., the quantity in the current year.

Correct Answer: ii) G.M. of Laspeyres and Paasche's

$$P_{\text{Fisher}} = \sqrt{P_L \times P_P} = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}} \times 100$$
It is the geometric mean (G.M.) of Laspeyres ($P_L$) and Paasche's ($P_P$) indices.

Correct Answer: iii) Price index

A price index measures the relative change in the price of a commodity (here, rice) over time. It is computed as:
$$\text{Price Index} = \frac{p_1}{p_0} \times 100$$

Correct Answer: iii) Consumer price index

The Consumer Price Index (CPI) measures the average change in prices paid by consumers for a basket of goods and services. The purchasing power of money is the reciprocal of CPI:
$$\text{Purchasing Power} = \frac{1}{\text{CPI}} \times 100$$