Kelly’s Price Index Calculator | केली मूल्य सूचकांक कैलकुलेटर

Kelly’s Price Index Calculator | केली मूल्य सूचकांक कैलकुलेटर

Kelly’s Price Index Calculator

केली मूल्य सूचकांक कैलकुलेटर

Calculate Kelly’s Price Index – weighted geometric mean of Laspeyres and Paasche indices. Get detailed step-by-step solutions.

Commodity P₀ (Base Price) P₁ (Current Price) Q₀ (Base Quantity) Q₁ (Current Quantity) Action
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Kelly’s Price Index Formula:

\[ P_{01}^{K} = \left( \frac{\sum P_1 Q_0}{\sum P_0 Q_0} \right)^{\alpha} \times \left( \frac{\sum P_1 Q_1}{\sum P_0 Q_1} \right)^{\beta} \times 100 \]

Where:
\( P_{01}^{K} \) = Kelly’s Price Index
\( \alpha \) = Weight for Laspeyres component (typically 0.5)
\( \beta \) = Weight for Paasche component (typically 0.5)
\( P_0 \) = Price in base year, \( P_1 \) = Price in current year
\( Q_0 \) = Quantity in base year, \( Q_1 \) = Quantity in current year

Kelly’s Index Calculation Results
Kelly’s Price Index
0.00
0.00% Change
α: 0.5, β: 0.5
Laspeyres Index (L)
0.00
0.00%
Paasche Index (P)
0.00
0.00%
Complete Calculation Table
Commodity P₀ P₁ Q₀ Q₁ P₀Q₀ P₁Q₀ P₀Q₁ P₁Q₁ Price Change %
← Scroll to view full calculation table →
Index Comparison Visualization
Step 1: Data Preparation
Step 2: Calculate Laspeyres Index (L)
Step 3: Calculate Paasche Index (P)
Step 4: Apply Kelly’s Formula
Step 5: Interpretation
Step 6: Detailed Analysis
Interpretation of Results
Kelly’s Price Index is a weighted geometric mean of Laspeyres and Paasche indices. It provides a flexible approach where different weights can be assigned to base and current year quantities based on their relative importance.

Kelly’s Price Index: Complete Guide

What is Kelly’s Price Index?

Kelly’s Price Index, developed by economist Tristram Kelly, is a weighted geometric mean of Laspeyres and Paasche price indices. This index allows for flexible weighting of base and current year quantities based on their relative importance or reliability.

Complete Formula and Calculation

Kelly’s Price Index Formula:

\[ P_{01}^{K} = \left( \frac{\sum P_1 Q_0}{\sum P_0 Q_0} \right)^{\alpha} \times \left( \frac{\sum P_1 Q_1}{\sum P_0 Q_1} \right)^{\beta} \times 100 \]

Where:

  • \( P_{01}^{K} \) = Kelly’s Price Index from period 0 to period 1
  • \( \alpha \) = Weight assigned to Laspeyres component (0 ≤ α ≤ 1)
  • \( \beta \) = Weight assigned to Paasche component (0 ≤ β ≤ 1)
  • Typically, α + β = 1, but not required
  • When α = β = 0.5, Kelly’s Index reduces to Fisher’s Ideal Index
  • \( P_0 \) = Price of each commodity in the base period
  • \( P_1 \) = Price of each commodity in the current period
  • \( Q_0 \) = Quantity of each commodity in the base period
  • \( Q_1 \) = Quantity of each commodity in the current period

Special Cases:

  • When α = 1, β = 0: Kelly’s = Laspeyres Index
  • When α = 0, β = 1: Kelly’s = Paasche Index
  • When α = β = 0.5: Kelly’s = Fisher’s Ideal Index

Advantages of Kelly’s Index

  • Flexible Weighting: Allows different weights for base and current periods
  • General Form: Includes Laspeyres, Paasche, and Fisher as special cases
  • Statistical Properties: Better statistical properties than arithmetic means
  • Balanced Approach: Can be tuned based on data reliability

Comparison with Other Indices

  • vs Fisher: Generalization of Fisher’s index with variable weights
  • vs Dorbish-Bowley: Uses geometric mean instead of arithmetic mean
  • vs Marshall-Edgeworth: Different mathematical approach but similar balanced perspective
  • vs Laspeyres/Paasche: More general form that includes both as special cases

Choosing Weights (α and β)

  • Equal Weights (α=β=0.5): When both periods’ data are equally reliable
  • α > β: When base year data is more reliable or important
  • α < β: When current year data is more reliable or important
  • Practical Approach: Weights can be based on sample sizes, data quality, or expert judgment

Frequently Asked Questions

How is Kelly’s Index different from Fisher’s Index?
Kelly’s Index is a generalization of Fisher’s Index. Fisher’s uses equal weights (α=β=0.5), while Kelly’s allows any weights for Laspeyres and Paasche components based on their relative importance.
How should I choose α and β weights?
Choose weights based on the reliability of data for each period. If base year data is more reliable, use higher α. If current year data is more reliable, use higher β. When both are equally reliable, use α=β=0.5.
Does Kelly’s Index satisfy time reversal test?
Only when α = β. In general, Kelly’s Index does not satisfy the time reversal test unless the weights are equal for both periods.
When is Kelly’s Index most useful?
Kelly’s Index is most useful when you have reason to believe that one period’s quantity data is more reliable or important than the other’s, and you want to reflect this in your price index calculation.

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