Study guide
4.3 Time Series Analysis & Index Numbers
ACCA Applied Knowledge — Management Accounting MA
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1 Chapter 1: Nature, Purpose & Source of Management Information
1.1 Role of Management Accounting, Planning Levels & Information Quality 1.2 Sources of Data, Big Data & Data Capture Costs 1.3 Cost Classification, Cost Behaviour & Coding Systems
2 Chapter 2: Cost Centres, Cost Units & Presenting Information
2.1 Cost Objects, Responsibility Centres & Managerial Needs 2.2 Presenting Management Information & Visualisation
3 Chapter 3: Sampling Methods, Statistical Summaries & Spreadsheets
3.1 Sampling Methods & Practical Selection 3.2 Data Summarisation, Dispersion & Probability in Decisions 3.3 Normal Distribution & Spreadsheet Applications in Accounting
4 Chapter 4: Forecasting Techniques, Time Series & Index Numbers
4.1 Linear Functions & The High-Low Method 4.2 Correlation, Linear Regression & Product Life Cycle 4.3 Time Series Analysis & Index Numbers
5 Chapter 5: Accounting for Materials & Inventory Control
5.1 Material Procedures, Physical Control & Inventory Valuation 5.2 Economic Order Quantity & Bulk Purchase Discounts 5.3 Economic Batch Quantity & Inventory Reorder Levels
6 Chapter 6: Accounting for Labour & Overhead Costs
6.1 Accounting for Labour, Remuneration Systems & Labour Ratios 6.2 Overhead Allocation, Apportionment & Reciprocal Reapportionment 6.3 Overhead Absorption Rates & Under/Over Absorption
7 Chapter 7: Marginal Costing & Absorption Costing
7.1 Marginal Costing, Contribution & Cost-Volume-Profit Analysis 7.2 Profit Comparison & Reconciliation Between Absorption and Marginal Costing
8 Chapter 8: Costing Methods & Alternative Principles
8.1 Job Costing, Batch Costing & Service/Operation Costing 8.2 Process Costing, Losses, Gains & Joint/By-Products 8.3 Alternative Cost Accounting: ABC, Target Costing & Life-Cycle Costing
9 Chapter 9: Budgetary Planning, Preparation & Behaviour
9.1 Budgetary Framework, Administration & The Principal Budget Factor 9.2 Functional Budgets, Cash Budgets & Master Budgets 9.3 What-If Analysis, Motivation & Behavioural Aspects of Budgeting
10 Chapter 10: Flexible Budgets & Capital Investment Appraisal
10.1 Flexible Budgets & Budgetary Control Reporting 10.2 Capital Expenditure, Compounding & Discounting Fundamentals 10.3 Discounted Cash Flow: NPV, IRR & Payback Methods
11 Chapter 11: Standard Costing & Basic Cost Variances
11.1 Standard Costing Purpose, Principles & Standard Cost Cards 11.2 Sales Price, Sales Volume & Direct Material Variances 11.3 Direct Labour & Overhead Variances
12 Chapter 12: Variance Investigation & Operating Statements
12.1 Variance Investigation, Interrelationships & Reverse Calculations 12.2 Operating Statements & Profit Reconciliations
13 Chapter 13: Performance Measurement Framework & Financial Analysis
13.1 Mission Statements, Objectives & Contextual Influences 13.2 Financial Performance: Profitability & Operating Efficiency Ratios 13.3 Financial Performance: Liquidity & Financial Gearing Ratios
14 Chapter 14: Divisional Performance, Multidimensional Models & Quality
14.1 Divisional Performance: Return on Investment & Residual Income 14.2 The Balanced Scorecard, Critical Success Factors & KPIs 14.3 Value for Money (3Es), Resource Utilisation & Non-Profit Performance 14.4 Quality Management, Benchmarking, Cost Control & Cost Reduction
4.3 Time Series Analysis & Index Numbers
Key Takeaways Explain This Topic AI
A time series decomposes historical observations into four distinct components: secular Trend (T), Seasonal variation (S), Cyclical variation (C), and Random or irregular variation (R).
Even-period moving averages (such as 4-quarter or 12-month series) require centering by calculating the average of successive moving average pairs to align trend estimates directly with actual calendar periods.
The additive model (Y = T + S + R) assumes seasonal variations are constant absolute monetary amounts where seasonal factors sum to zero, whereas the multiplicative model (Y = T × S × R) assumes seasonal variations scale proportionally with the trend.
Index numbers track relative changes in price or quantity over time against a chosen base period (set to 100); multi-item weighted indices are essential to reflect the relative economic importance of individual commodities.
The Laspeyres index utilizes base-period weights (Q0) and tends to overstate inflation by ignoring the substitution effect, whereas the Paasche index uses current-period weights (Qn) and tends to understate inflation.
Last updated: September 2026
4.3 Time Series Analysis & Index Numbers
In operational budgeting and resource planning, management accountants must forecast revenues and costs that fluctuate systematically over calendar periods. For example, hotel bookings peak during summer holidays, toy manufacturers experience surges prior to festive seasons, and agricultural processing costs surge during harvest months. To isolate underlying commercial trends from predictable seasonal cycles and unpredictable external shocks, organizations employ time series analysis . Furthermore, to manage the eroding impact of price changes across complex purchasing baskets, management accountants utilize index numbers .
Advantages and limitations of time-series analysis
Advantages: It converts historical observations into an objective, repeatable forecasting base; separates trend and seasonality so budgets can reflect recurring calendar patterns; and makes changes over time visible for planning, staffing, inventory, and control.
Limitations: It assumes past patterns remain relevant, so structural breaks, new competitors, regulation, or one-off shocks can invalidate a forecast. Results also depend on adequate, consistent data and informed selection of the model. Moving averages lag turning points and lose observations at the ends of a series, while an apparent historical relationship does not by itself establish causation. Management should therefore combine the statistical forecast with current commercial evidence and sensitivity analysis.
1. The Four Components of a Time Series
A time series is a chronological sequence of numerical observations recorded at equal, successive time intervals (e.g., hourly energy demand, weekly warehouse throughput, monthly departmental overhead, or quarterly sales revenue). Statistical decomposition breaks a time series into four constituent forces: Actual Sales ($)
- ^
- | / Trend Line (T)
- | /\ / (Long-Term Direction)
- | / \ /\ /
- | /\ / \ / \ /
- | / \ /\ / \/ \ /
- | / \/ \/ \ /
- | / \/ <--- Actual Time Series (Y)
- | / Incorporates: T + S + C + R
- 0 +----+-------------------------------------------------------------> Time
Trend ( T T T ): The underlying, long-term secular direction of the data over an extended timeframe. The trend reflects persistent, fundamental momentum—such as steady business expansion, gradual population growth, technological adoption, or sustained industry decline. Trend lines may be upward-sloping, downward-sloping, or stationary.
Seasonal Variation ( S S S ): Short-term, regular, periodic fluctuations that repeat systematically within a fixed, known timeframe (most commonly within a single calendar year, but also within a weekly or 24-hour cycle). Seasonality is driven by recurring calendar events: weather seasons (winter heating, summer tourism), public holidays (Christmas retail shopping), or administrative deadlines (quarterly tax filings).
Cyclical Variation ( C C C ): Medium-to-long-term, wave-like oscillations around the secular trend driven by broader macroeconomic business cycles (economic expansions, booms, recessions, and recoveries). Unlike seasonal cycles, cyclical variations do not repeat at fixed calendar intervals; a macroeconomic cycle typically spans three to ten years. In short-to-medium-term operational budgeting (ACCA MA level), cyclical variations are generally subsumed into the long-term trend.
Random / Irregular Variation ( R R R ): Unpredictable, erratic residual fluctuations caused by unforeseen, non-recurring external disturbances (e.g., industrial strikes, sudden geopolitical conflicts, extreme natural disasters, or unexpected software outages). Random noise cannot be forecasted mathematically.
2. Time Series Decomposition Models: Additive vs. Multiplicative
To recombine these components into a forecasting model, management accountants use either an additive model or a multiplicative (proportional) model . Additive Model (Constant Swings) Multiplicative Model (Proportional Swings)
- ^ ^
- | /\ | /\
- | /\ / \ /\ | / \
- | / \ / \ / \ | /\ / \
- | / \ / \ / \ | /\ / \ / \
- | / V V \ | / \ / \ / \
- | / \ | / \ / \ / \
- |/ Trend Line (T) \ | / V \ / \
- -+----------------------------> Time --+------------------V/---------------> Time
- Seasonal swings remain constant Seasonal swings expand as Trend rises
- in absolute dollar terms in fixed percentage proportion
- A. The Additive Model
The additive model assumes that the components act independently, and seasonal variations represent constant absolute amounts added to or subtracted from the trend: Y = T + S + R Y = T + S + R Y = T + S + R
Measurement Units: Seasonal variations are expressed in absolute financial or physical units (e.g., + $ 15 , 000 +\text{\textdollar}15,000 + $ 15 , 000 in Quarter 3, − $ 8 , 000 -\text{\textdollar}8,000 − $ 8 , 000 in Quarter 1).
Mathematical Constraint: Over a complete seasonal cycle (e.g., 4 quarters or 12 months), the sum of the seasonal variations must equal zero: ∑ S = 0 \sum S = 0 ∑ S = 0
If the raw calculated seasonal variations do not sum to zero due to random rounding or noise, an adjustment is applied: calculate the average error ∑ S k \frac{\sum S}{k} k ∑ S (where k k k is the number of periods in the cycle) and subtract this correction from each individual seasonal component.
B. The Multiplicative (Proportional) Model
The multiplicative model assumes that seasonal variations are proportional to the trend . As the baseline trend increases, the magnitude of seasonal peaks and troughs expands proportionally: Y = T × S × R or Y = T × S Y = T \times S \times R \quad \text{or} \quad Y = T \times S Y = T × S × R or Y = T × S
Measurement Units: Seasonal variations are expressed as ratios, proportions, or percentages of the trend (e.g., 1.25 1.25 1.25 or 125 % 125\% 125% in Quarter 3, 0.80 0.80 0.80 or 80 % 80\% 80% in Quarter 1).
Mathematical Constraint: Across a complete seasonal cycle, the average seasonal factor must equal 1.00 1.00 1.00 (or 100 % 100\% 100% ). Thus, the sum of the factors must equal the number of periods in the cycle: ∑ S = k ( 4.00 for quarterly data; 12.00 for monthly data ) \sum S = k \quad (4.00 \text{ for quarterly data; } 12.00 \text{ for monthly data}) ∑ S = k ( 4.00 for quarterly data; 12.00 for monthly data )
If the raw factors sum to a value different from k k k , normalize them by multiplying each factor by k ∑ S raw \frac{k}{\sum S_{\text{raw}}} ∑ S raw k .
- Feature Additive Model Multiplicative Model
- Equation Y = T + S + R Y = T + S + R Y = T + S + R Y = T × S × R Y = T \times S \times R Y = T × S × R
- Nature of Seasonality Absolute fixed monetary/physical amount Relative percentage/ratio of the trend
Behavior over Time Seasonal swings remain constant in size regardless of trend growth Seasonal swings grow or shrink in direct proportion to trend growth
Mathematical Baseline ∑ S = 0 \sum S = 0 ∑ S = 0 (Sum equals zero) ∑ S = 4.0 \sum S = 4.0 ∑ S = 4.0 (quarters) or 12.0 12.0 12.0 (months); Mean S = 1.0 S = 1.0 S = 1.0
Best Application Stable, mature markets with zero or slow secular growth Growing or contracting markets where volume swings scale with size
3. Calculating the Trend via Moving Averages
The primary method used to eliminate seasonal and random fluctuations and extract the underlying secular trend ( T T T ) is the moving average technique .
Odd-Period Moving Averages (e.g., 3-Month or 5-Quarter)
When smoothing across an odd number of time periods (e.g., 5 quarters), the arithmetic mean aligns naturally with the middle time period. For example, the average of periods 1, 2, 3, 4, and 5 centers squarely on Period 3 .
Even-Period Moving Averages & The Centering Technique (e.g., 4-Quarter or 12-Month)
In business forecasting, data is most frequently organized into 4 quarters or 12 months. Calculating an average across an even number of periods creates an alignment dilemma:
The average of Quarters 1, 2, 3, and 4 falls at the midpoint between Quarter 2 and Quarter 3 ( t = 2.5 t = 2.5 t = 2.5 ).
Because no actual calendar observation exists at t = 2.5 t = 2.5 t = 2.5 , this moving average cannot be compared directly against any recorded historical figure.
To resolve this, management accountants calculate a centered moving average :
Compute 4-quarter moving totals across the series.
Compute a second 2-period moving average of the 4-quarter moving totals (or sum adjacent pairs of 4-quarter totals and divide by 4 × 2 = 8 4 \times 2 = 8 4 × 2 = 8 ).
This shifts the mathematical center from 2.5 2.5 2.5 directly onto Quarter 3 , creating an aligned trend estimate ( T T T ).
- 4. Comprehensive Worked Example: Moving Averages, Seasonality & Forecasting
- Historical Data
Highland Distillers Ltd records quarterly sales volume (thousands of cases) over two consecutive operating years:
- Year Quarter Sales Volume, Y Y Y ( ′ 000 '000 ′ 000 )
- Year 1 Q1 120
- Q2 180
- Q3 240
- Q4 140
- Year 2 Q1 150
- Q2 210
- Q3 280
- Q4 170
Step-by-Step Trend Calculation via Centered Moving Averages +-------------------------------------------------------------------------------------------------------+
- | Year/Qtr | Actual Y | 4-Qtr Moving Total | 8-Qtr Centered Total | Centered Trend (T) | Additive S = Y - T |
- +----------+----------+--------------------+----------------------+--------------------+--------------------+
- | Y1 Q1 | 120 | -- | -- | -- | -- |
- | Y1 Q2 | 180 | | | | |
- | | | (120+180+240+140) | | | |
- | | | = 680 | | | |
- | | | | (680 + 710) | 1,390 / 8 | 240 - 173.75 |
- | Y1 Q3 | 240 | | = 1,390 | = 173.75 | = +66.25 |
- | | | (180+240+140+150) | | | |
- | | | = 710 | | | |
- | | | | (710 + 740) | 1,450 / 8 | 140 - 181.25 |
- | Y1 Q4 | 140 | | = 1,450 | = 181.25 | = -41.25 |
- | | | (240+140+150+210) | | | |
- | | | = 740 | | | |
- | | | | (740 + 780) | 1,520 / 8 | 150 - 190.00 |
- | Y2 Q1 | 150 | | = 1,520 | = 190.00 | = -40.00 |
- | | | (140+150+210+280) | | | |
- | | | = 780 | | | |
- | | | | (780 + 810) | 1,590 / 8 | 210 - 198.75 |
- | Y2 Q2 | 210 | | = 1,590 | = 198.75 | = +11.25 |
- | | | (150+210+280+170) | | | |
- | | | = 810 | | | |
- | Y2 Q3 | 280 | -- | -- | -- | -- |
- | Y2 Q4 | 170 | -- | -- | -- | -- |
- +-------------------------------------------------------------------------------------------------------+
- Step-by-Step Trend Line Modeling
Inspect the centered trend values across successive quarters:
Y1 Q3: 173.75 173.75 173.75
Y1 Q4: 181.25 181.25 181.25 (Quarterly increase = 181.25 − 173.75 = 7.50 = 181.25 - 173.75 = 7.50 = 181.25 − 173.75 = 7.50 )
Y2 Q1: 190.00 190.00 190.00 (Quarterly increase = 190.00 − 181.25 = 8.75 = 190.00 - 181.25 = 8.75 = 190.00 − 181.25 = 8.75 )
Y2 Q2: 198.75 198.75 198.75 (Quarterly increase = 198.75 − 190.00 = 8.75 = 198.75 - 190.00 = 8.75 = 198.75 − 190.00 = 8.75 )
Average quarterly trend increase: Δ T = 198.75 − 173.75 3 quarters = 25.00 3 ≈ 8.33 thousand cases per quarter \Delta T = \frac{198.75 - 173.75}{3 \text{ quarters}} = \frac{25.00}{3} \approx 8.33 \text{ thousand cases per quarter} Δ T = 3 quarters 198.75 − 173.75 = 3 25.00 ≈ 8.33 thousand cases per quarter
Establishing Seasonal Variations (Additive Model)
From our calculations:
- Q1 Variation (Y2 Q1): − 40.00 -40.00 − 40.00
- Q2 Variation (Y2 Q2): + 11.25 +11.25 + 11.25
- Q3 Variation (Y1 Q3): + 66.25 +66.25 + 66.25
- Q4 Variation (Y1 Q4): − 41.25 -41.25 − 41.25
Check summation condition: ∑ S = ( − 40.00 ) + ( + 11.25 ) + ( + 66.25 ) + ( − 41.25 ) = − 3.75 \sum S = (-40.00) + (+11.25) + (+66.25) + (-41.25) = -3.75 ∑ S = ( − 40.00 ) + ( + 11.25 ) + ( + 66.25 ) + ( − 41.25 ) = − 3.75
Because the sum does not equal zero, apply the standard ACCA normalization adjustment: Adjustment per quarter = − 3.75 4 = − 0.9375 \text{Adjustment per quarter} = \frac{-3.75}{4} = -0.9375 Adjustment per quarter = 4 − 3.75 = − 0.9375
Subtract − 0.9375 -0.9375 − 0.9375 (i.e., add 0.9375 0.9375 0.9375 ) to each quarter:
Adjusted Q1: − 40.00 − ( − 0.9375 ) = − 39.0625 ≈ − 39.06 -40.00 - (-0.9375) = -39.0625 \approx -39.06 − 40.00 − ( − 0.9375 ) = − 39.0625 ≈ − 39.06
Adjusted Q2: + 11.25 − ( − 0.9375 ) = + 12.1875 ≈ + 12.19 +11.25 - (-0.9375) = +12.1875 \approx +12.19 + 11.25 − ( − 0.9375 ) = + 12.1875 ≈ + 12.19
Adjusted Q3: + 66.25 − ( − 0.9375 ) = + 67.1875 ≈ + 67.19 +66.25 - (-0.9375) = +67.1875 \approx +67.19 + 66.25 − ( − 0.9375 ) = + 67.1875 ≈ + 67.19
Adjusted Q4: − 41.25 − ( − 0.9375 ) = − 40.3125 ≈ − 40.31 -41.25 - (-0.9375) = -40.3125 \approx -40.31 − 41.25 − ( − 0.9375 ) = − 40.3125 ≈ − 40.31
Check: ( − 39.06 ) + ( + 12.19 ) + ( + 67.19 ) + ( − 40.31 ) = 0.00 (-39.06) + (+12.19) + (+67.19) + (-40.31) = 0.00 ( − 39.06 ) + ( + 12.19 ) + ( + 67.19 ) + ( − 40.31 ) = 0.00 (Satisfied).
Forecasting for Year 3 Quarter 1 (Additive Model)
Starting from Y2 Q2 ( T = 198.75 T = 198.75 T = 198.75 at t = 6 t=6 t = 6 ):
Year 3 Quarter 1 corresponds to t = 9 t = 9 t = 9 (3 quarters beyond Y2 Q2): Projected Trend T Y3 Q1 = 198.75 + ( 3 × 8.33 ) = 198.75 + 24.99 = 223.74 \text{Projected Trend } T_{\text{Y3 Q1}} = 198.75 + (3 \times 8.33) = 198.75 + 24.99 = 223.74 Projected Trend T Y3 Q1 = 198.75 + ( 3 × 8.33 ) = 198.75 + 24.99 = 223.74 Forecasted Sales Y = T + S Q1 = 223.74 + ( − 39.06 ) = 184.68 thousand cases \text{Forecasted Sales } Y = T + S_{\text{Q1}} = 223.74 + (-39.06) = 184.68 \text{ thousand cases} Forecasted Sales Y = T + S Q1 = 223.74 + ( − 39.06 ) = 184.68 thousand cases
5. Index Numbers: Purpose and Conceptual Foundations
Management accountants operate across dynamic economic environments where the purchasing power of money shifts constantly. An index number is a statistical ratio designed to measure relative changes in price, cost, quantity, or volume over time, compared against an established benchmark known as the base period .
Fundamentals
Base Period ( 0 0 0 ): The reference period against which all other periods are evaluated. The base period is conventionally assigned an index value of 100 .
Current Period ( n n n ): The period being measured.
Simple Price and Quantity Indices
For a single commodity:
Simple Price Index ( I P I_P I P ): I P = P n P 0 × 100 I_P = \frac{P_n}{P_0} \times 100 I P = P 0 P n × 100
Simple Quantity Index ( I Q I_Q I Q ): I Q = Q n Q 0 × 100 I_Q = \frac{Q_n}{Q_0} \times 100 I Q = Q 0 Q n × 100
Where P 0 , Q 0 P_0, Q_0 P 0 , Q 0 represent price and quantity in the base period, and P n , Q n P_n, Q_n P n , Q n represent price and quantity in the current period.
Limitation of Simple Unweighted Indices
In reality, organizations purchase hundreds of raw materials and operational services. Simple unweighted arithmetic averages treat all items equally: a 50% price surge in office envelopes would be given the exact same weight as a 50% surge in raw steel sheet procurement. To produce meaningful managerial information, multi-item indices must be weighted according to economic importance (expenditure or volume).
6. Multi-Item Weighted Indices: Laspeyres vs. Paasche
The two primary multi-item index number formulations tested in the ACCA syllabus are the Laspeyres Index and the Paasche Index .
A. The Laspeyres Index (Base-Period Weighted)
The Laspeyres index weights price or quantity changes using base-period quantities ( Q 0 Q_0 Q 0 ) or base-period prices ( P 0 P_0 P 0 ) :
Laspeyres Price Index ( L P L_P L P ): L P = ∑ P n Q 0 ∑ P 0 Q 0 × 100 L_P = \frac{\sum P_n Q_0}{\sum P_0 Q_0} \times 100 L P = ∑ P 0 Q 0 ∑ P n Q 0 × 100
Laspeyres Quantity Index ( L Q L_Q L Q ): L Q = ∑ P 0 Q n ∑ P 0 Q 0 × 100 L_Q = \frac{\sum P_0 Q_n}{\sum P_0 Q_0} \times 100 L Q = ∑ P 0 Q 0 ∑ P 0 Q n × 100
B. The Paasche Index (Current-Period Weighted)
The Paasche index weights price or quantity changes using current-period quantities ( Q n Q_n Q n ) or current-period prices ( P n P_n P n ) :
Paasche Price Index ( P P P_P P P ): P P = ∑ P n Q n ∑ P 0 Q n × 100 P_P = \frac{\sum P_n Q_n}{\sum P_0 Q_n} \times 100 P P = ∑ P 0 Q n ∑ P n Q n × 100
Paasche Quantity Index ( P Q P_Q P Q ): P Q = ∑ P n Q n ∑ P n Q 0 × 100 P_Q = \frac{\sum P_n Q_n}{\sum P_n Q_0} \times 100 P Q = ∑ P n Q 0 ∑ P n Q n × 100
7. Operational Evaluation and Economic Biases: Laspeyres vs. Paasche +---------------------------------------------------------------------------------------+
- | INDEX NUMBER MATRIX |
- +-----------------------+----------------------------------+----------------------------+
- | Dimension | Laspeyres Index (Base Weighted) | Paasche Index (Current Wt) |
- +-----------------------+----------------------------------+----------------------------+
- | Weighting Mechanism | Fixed Base-Period Basket (Q0) | Changing Current Basket(Qn)|
- +-----------------------+----------------------------------+----------------------------+
- | Computational Cost | Inexpensive and fast. | Expensive and slow. |
- | | Quantities surveyed only ONCE. | Requires surveying fresh |
- | | Only current prices Pn needed. | quantities Qn each period. |
- +-----------------------+----------------------------------+----------------------------+
- | Inter-Year Comparison | Valid across multiple years: | Invalid across years: |
- | | Basket remains strictly constant.| Weights change every year. |
- +-----------------------+----------------------------------+----------------------------+
- | Inherent Economic | OVERSTATES Inflation: | UNDERSTATES Inflation: |
- | Bias | Ignores the substitution effect; | Over-weights substituted |
- | | assumes consumers buy expensive | cheaper goods; understates |
- | | goods in identical quantities. | true cost-of-living rise. |
- +-----------------------+----------------------------------+----------------------------+
- The Substitution Effect and Respective Biases
Why Laspeyres Overstates Inflation ( L P L_P L P ): In commercial reality, when the price of one raw material escalates dramatically, rational procurement managers substitute away from it toward cheaper alternatives. Because the Laspeyres index holds base-period consumption quantities ( Q 0 Q_0 Q 0 ) rigidly fixed, it assumes the firm continues purchasing the expensive item in unaltered quantities, thereby overstating the true cost increase .
Why Paasche Understates Inflation ( P P P_P P P ): The Paasche index uses current-period quantities ( Q n Q_n Q n ), which already reflect management's defensive substitution away from inflated commodities into lower-cost substitutes. Consequently, the current basket under-represents items that suffered the highest price increases, systematically understating the true rate of inflation .
- 8. Fully Worked Example: Laspeyres vs. Paasche Price Indices
- Purchasing Basket Data
A chemical processor purchases three core industrial ingredients:
Material Base Year Price, P 0 P_0 P 0 Base Year Qty, Q 0 Q_0 Q 0 Current Year Price, P 1 P_1 P 1 Current Year Qty, Q 1 Q_1 Q 1
- Alpha $10 50 $12 60
- Beta $20 30 $30 20
- Gamma $40 10 $35 12
Notice that Beta experienced a massive 50% price increase (from $20 to $30), causing procurement to reduce consumption from 30 to 20 units (substitution effect).
Tabulation of Cross-Products
Material P 0 Q 0 P_0 Q_0 P 0 Q 0 P 1 Q 0 P_1 Q_0 P 1 Q 0 P 1 Q 1 P_1 Q_1 P 1 Q 1 P 0 Q 1 P_0 Q_1 P 0 Q 1
Alpha 10 × 50 = 500 10 \times 50 = 500 10 × 50 = 500 12 × 50 = 600 12 \times 50 = 600 12 × 50 = 600 12 × 60 = 720 12 \times 60 = 720 12 × 60 = 720 10 × 60 = 600 10 \times 60 = 600 10 × 60 = 600
Beta 20 × 30 = 600 20 \times 30 = 600 20 × 30 = 600 30 × 30 = 900 30 \times 30 = 900 30 × 30 = 900 30 × 20 = 600 30 \times 20 = 600 30 × 20 = 600 20 × 20 = 400 20 \times 20 = 400 20 × 20 = 400
Gamma 40 × 10 = 400 40 \times 10 = 400 40 × 10 = 400 35 × 10 = 350 35 \times 10 = 350 35 × 10 = 350 35 × 12 = 420 35 \times 12 = 420 35 × 12 = 420 40 × 12 = 480 40 \times 12 = 480 40 × 12 = 480
Total ∑ P 0 Q 0 = 1 , 500 \sum P_0 Q_0 = 1,500 ∑ P 0 Q 0 = 1 , 500 ∑ P 1 Q 0 = 1 , 850 \sum P_1 Q_0 = 1,850 ∑ P 1 Q 0 = 1 , 850 ∑ P 1 Q 1 = 1 , 740 \sum P_1 Q_1 = 1,740 ∑ P 1 Q 1 = 1 , 740 ∑ P 0 Q 1 = 1 , 480 \sum P_0 Q_1 = 1,480 ∑ P 0 Q 1 = 1 , 480
Index Calculations
1. Laspeyres Price Index ( L P L_P L P ): L P = ∑ P 1 Q 0 ∑ P 0 Q 0 × 100 = 1 , 850 1 , 500 × 100 = 123.33 L_P = \frac{\sum P_1 Q_0}{\sum P_0 Q_0} \times 100 = \frac{1,850}{1,500} \times 100 = 123.33 L P = ∑ P 0 Q 0 ∑ P 1 Q 0 × 100 = 1 , 500 1 , 850 × 100 = 123.33
Indicates an estimated price inflation of +23.33% .
2. Paasche Price Index ( P P P_P P P ): P P = ∑ P 1 Q 1 ∑ P 0 Q 1 × 100 = 1 , 740 1 , 480 × 100 = 117.57 P_P = \frac{\sum P_1 Q_1}{\sum P_0 Q_1} \times 100 = \frac{1,740}{1,480} \times 100 = 117.57 P P = ∑ P 0 Q 1 ∑ P 1 Q 1 × 100 = 1 , 480 1 , 740 × 100 = 117.57
Indicates an estimated price inflation of +17.57% .
Analytical Conclusion
As predicted by economic theory, L P ( 123.33 ) > P P ( 117.57 ) L_P (123.33) > P_P (117.57) L P ( 123.33 ) > P P ( 117.57 ) . The Laspeyres index overstates cost increases because it weights Material Beta at its original high volume of 30 units, whereas the Paasche index captures the cost-saving shift to 20 units.