Passche’s and Fisher’s Index numbers

Paasche’s Index is another method used to calculate price and quantity index numbers, named after the German economist Hermann Paasche. Unlike the Laspeyres’ Index, which uses base period quantities as weights, the Paasche’s Index uses the current period quantities as weights. This approach makes the Paasche’s Index more reflective of actual consumption or production patterns in the current period, as it takes into account the quantities of goods or services that are actually being consumed or produced in the current period.

Formula for Paasche’s Price Index:

PP = [∑(Pt × Qt) / ∑(P0 × Qt)] × 100

Where:

• Pt​ = Price of the commodity in the current period
• P0​ = Price of the commodity in the base period
• Qt​ = Quantity of the commodity in the current period
• Pp​ = Paasche’s Price Index

Formula for Paasche’s Quantity Index:

QP = [ ∑(Pt×Qt) / ∑(P0×Q0) ] × 100

Where:

• Qp​ = Paasche’s Quantity Index
• Q0​ = Quantity of the commodity in the base period

Key Characteristics of Paasche’s Index:

1. Current Period Weights: Paasche’s Index uses current period quantities as weights. This method adapts to changes in consumption and production patterns over time.
2. Reflections of Current Consumption: Paasche’s Index is more accurate in reflecting the true changes in the cost of living or production because it takes current consumption into account. If consumers have changed their buying habits due to price changes, the Paasche Index will reflect these changes.
3. Understating Price or Quantity Changes: Since Paasche’s Index uses current quantities, it may underestimate price or quantity changes in comparison to Laspeyres’ Index. This happens because, if prices have risen, consumers may shift toward cheaper goods, thus reducing the relative weight of expensive items in the current period.

Example of Paasche’s Price Index:

Let’s use the same commodities as in the Laspeyres’ example:

To calculate the Paasche’s Price Index:

Pp = [ (15×6)+(25×12)+(40×10) / (10×6)+(20×12)+(30×10)] × 100

Numerator (current period prices weighted by current period quantities):

= (15×6) + (25×12) + (40×10) = 90 + 300 + 400 = 790

Denominator (base period prices weighted by current period quantities):

= (10×6) + (20×12) + (30×10) = 60 + 240 + 300 = 600

Now, substitute into the formula:

Pp = [790 / 600] × 100 = 131.67

This indicates a 31.67% increase in prices relative to the base period.

Fisher’s Index Number:

Fisher’s Index is a combination of both Laspeyres’ Index and Paasche’s Index. Named after the economist Irving Fisher, this index is considered a “ideal” index because it takes the geometric mean of the Laspeyres’ and Paasche’s indexes, which balances the overstatement and understatement of price or quantity changes inherent in the two individual methods.

Fisher Index is also known as the Fisher Ideal Index because it attempts to address the shortcomings of both Laspeyres’ and Paasche’s indices by providing a more accurate measure of price or quantity changes.

Formula for Fisher’s Price Index:

FP = √Lp × Pp

​Where:

• Lp​ = Laspeyres’ Price Index
• Pp​ = Paasche’s Price Index

Formula for Fisher’s Quantity Index:

FQ = √(Lq x Pq)

​Where:

• Lq​ = Laspeyres’ Quantity Index
• Pq​ = Paasche’s Quantity Index

Key Characteristics of Fisher’s Index:

1. Balanced Approach: Fisher’s Index uses both the Laspeyres’ and Paasche’s indices to create a balanced measure that avoids the biases of using only one of the two approaches.
2. Geometric Mean: By taking the geometric mean of the Laspeyres and Paasche indices, Fisher’s Index smooths out the extreme effects that might arise from the rigid base period quantities (Laspeyres) or the substitution effect (Paasche).
3. Ideal Index: Fisher’s Index is widely regarded as the most accurate index number for price and quantity comparisons because it combines the advantages of both other indices.

Example of Fisher’s Price Index:

Using the Laspeyres’ Price Index from the previous example (131.37) and the Paasche’s Price Index (131.67):

FP = √(131.37×131.67) = √(17338.82( ≈ 131.52

This result, 131.52, indicates a price increase of approximately 31.52% compared to the base period.

Advantages:

1. Balanced Calculation: Fisher’s Index combines the advantages of both Laspeyres and Paasche indices, offering a more reliable measure of price and quantity changes.
2. Widely Accepted: It is widely accepted in academic and practical applications due to its balanced nature.

Disadvantages:

1. Complex Calculation: It is more complex to compute than the Laspeyres and Paasche indices, as it requires calculating both indices and then finding the geometric mean.
2. Data Intensive: Fisher’s Index requires data from both periods (base and current) to calculate the price and quantity changes.