Unit 5: Detailed Study Notes on Statistical Index Numbers

Overview and Learning Outcomes of Index Numbers

• Definition: An index number measures the change in a variable over time relative to the value of the variable during a pre-selected base period. It is a relative figure expressed as a percentage.

• Business Application: Business forecasts are based on historical data. Index numbers are used in business and economics as indicators to measure how much an economic variable changes over time or differs between two locations.

• Core Objectives:     * Calculate simple index numbers.     * Calculate composite index numbers.     * Distinguish between unweighted and weighted index numbers.     * Change the base period for a data series.     * Calculate Link relatives.     * Understand the mechanics of the Consumer Price Index (CPI).

Fundamental Concepts of Calculation

• Calculation Process: An index number is calculated by dividing the current value (numerator) by a base value (denominator) and multiplying the result by 100.

• The Base Period: This is the specific point in time to which the comparison is made.

• Base Year Index: The index for the base year is always $100$.

• Interpretation of Results:     * Subsequent years will be above or below $100$ depending on whether there has been an increase or decrease relative to the base year.     * Example: An index number of $124$ signifies a $24\%$ increase.     * Example: An index number of $93$ signifies a $7\%$ decrease.

• Selecting a Base Period: Ideally, the period should be:     * Recent enough so that changing technology and consumer behavior do not affect comparisons.     * A "typical" period regarding the activity of interest.     * Consistent with the periods used by other series intended for comparison.     * A period of economic stability free from abnormal influences.

Classification of Index Numbers

• Variables Measured:     * Price Index ($I_p$): Compares changes in prices from one period to another.     * Quantity/Volume Index ($I_q$): Measures how much the quantity of a variable changes over time.     * Value Index: Measures changes in total monetary worth by combining price and quantity components.

• Scope of Items:     * Simple Index Number: Represents a comparison for a single individual item.     * Aggregate or Composite Index Number: Constructed for a group of items, often referred to as a "basket of goods."

• Weighting Methods:     * Unweighted Index Numbers: Deal only with price or quantity in the calculation, treating all items as equally important.     * Weighted Index Numbers: Factor in both price and quantity to reflect the relative importance of different components in the calculation.

Simple Index Number Calculations

• General Formula for Simple Index ($I$):     $I = \frac{\text{Current year data}}{\text{Base year data}} \times 100$

• Calculation Steps:     1. Select the base period.     2. Divide the current value ($P_i$ or $Q_i$) by the base value ($P_b$ or $Q_b$).     3. Multiply the ratio by $100$.

• Example 5.1 (Milk Price):     * Price 1999: $R3.50/liter$ (Base)     * Price 2000: $R3.85/liter$ (Current)     * Calculation: $\frac{3.85}{3.50} \times 100 = 110$     * Interpretation: The price of milk increased by $10\%$

• Activity 5.1 (Simple Index per Commodity):     * Peanuts:         * Price: $1998 = R45$, $1999 = R50$. Index: $\frac{50}{45} \times 100 = 111.11$ ($11.11\%$ increase).         * Quantity: $1998 = 600$, $1999 = 550$. Index: $\frac{550}{600} \times 100 = 91.67$ ($8.33\%$ decrease).     * Pecans:         * Price: $1998 = R55$, $1999 = R60$. Index: $\frac{60}{55} \times 100 = 109.09$ ($9.09\%$ increase).         * Quantity: $1998 = 300$, $1999 = 350$. Index: $\frac{350}{300} \times 100 = 116.67$ ($16.67\%$ increase).     * Cashews:         * Price: $1998 = R60$, $1999 = R70$. Index: $\frac{70}{60} \times 100 = 116.67$ ($16.67\%$ increase).         * Quantity: $1998 = 325$, $1999 = 325$. Index: $\frac{325}{325} \times 100 = 100.00$ (No change).

Construction of Aggregate (Composite) Index Numbers

• Unweighted Aggregate Index: Used when every item in the basket is considered of equal importance.     * Formula for Price Index ($I_p$): $I_p = \frac{\sum P_i}{\sum P_b} \times 100$     * Steps: Sum the prices for all items in the current period ($\sum P_i$), sum the prices for all items in the base period ($\sum P_b$), divide the sums, and multiply by $100$.

• Example 5.2 (Student Materials):     * Items: Textbook ($2007: 203, 2008: 229$), Calculator ($2007: 80, 2008: 70$), Answer Manual ($2007: 10, 2008: 10$).     * Sum of 2007 Prices ($\sum P_b$): $293$     * Sum of 2008 Prices ($\sum P_i$): $309$     * Index: $\frac{309}{293} \times 100 = 105.46$     * Interpretation: Prices increased by $5.46\%$

• Weighted Composite Index: Applied when items have different levels of importance. The price of an item is typically weighted by the quantity sold.     * Laspeyres' Index: Uses quantities consumed during the base period as the weighting factor.         * Assumption: Quantities remain constant regardless of price changes.         * Formula: $I_p(\text{Laspeyres}) = \frac{\sum P_i Q_b}{\sum P_b Q_b} \times 100$         * Advantages: Quantitative consumption data only needs to be measured during the base year.         * Disadvantages: It does not account for changes in consumption patterns over time.     * Paasche's Index: Uses quantities consumed during the current period as the weighting factor.         * Formula: $I_p(\text{Paasche}) = \frac{\sum P_i Q_i}{\sum P_b Q_i} \times 100$         * Purpose: Measures change based on current year consumption patterns, avoiding issues with outdated patterns.

Weighted Index Case Studies (Example 5.3)

• Data Table for Orchard Production (2007 Base):     * Apples: $P_b=20, P_i=25, Q_b=2.0, Q_i=1.5$     * Oranges: $P_b=15, P_i=17, Q_b=1.4, Q_i=1.6$     * Mangos: $P_b=40, P_i=35, Q_b=5.0, Q_i=2.0$     * Bananas: $P_b=30, P_i=38, Q_b=7.0, Q_i=9.0$

• Laspeyres Calculation:     * $\sum P_i Q_b = (25 \times 2.0) + (17 \times 1.4) + (35 \times 5.0) + (38 \times 7.0) = 50.0 + 23.8 + 175.0 + 266.0 = 514.8$     * $\sum P_b Q_b = (20 \times 2.0) + (15 \times 1.4) + (40 \times 5.0) + (30 \times 7.0) = 40.0 + 21.0 + 200.0 + 210.0 = 471.0$     * Index: $\frac{514.8}{471.0} \times 100 = 109.3$     * Result: $9.30\%$ price increase.

• Paasche Calculation:     * $\sum P_i Q_i = (25 \times 1.5) + (17 \times 1.6) + (35 \times 2.0) + (38 \times 9.0) = 37.5 + 27.2 + 70.0 + 342.0 = 476.7$     * $\sum P_b Q_i = (20 \times 1.5) + (15 \times 1.6) + (40 \times 2.0) + (30 \times 9.0) = 30.0 + 24.0 + 80.0 + 270.0 = 404.0$     * Index: $\frac{476.7}{404.0} \times 100 = 118.0$     * Result: Approximately $18\%$ price increase (Slide 33 notes activity result as $112$ or $12\%$ for different activity data).

The Consumer Price Index (CPI)

• Significance: A vital economic indicator used to determine inflation rates and the cost of living.

• Authority: Published monthly by the Department of Statistics.

• Methodology in South Africa: Uses the Laspeyres formula:     $\text{CPI} = \frac{\sum P_i Q_b}{\sum P_b Q_b} \times 100$

• Establishment of Weights: Weight factors are determined by sampling at least $10,000$ households across various income groups and metropolitan areas.

• Update Frequency: International practice dictates the base period for CPI must change at least every 5 years.

• Calculation of Inflation Rate:     $\text{Inflation Rate} = \frac{\text{CPI current year}}{\text{CPI of corresponding month of previous year}}$

Advanced Index Number Properties

• Percentage Points Change: Subtracting one index from another determines "percentage points." As indices grow larger, the same percentage change corresponds to a larger numerical difference in points.     * Example 5.4:         * Index $100$ to $120$: $20$ points ($20\%$ increase).         * Index $120$ to $160$: $40$ points ($\frac{160-120}{120} \times 100 = 33.33\%$ increase).

• Link Relatives: Indexes where the base is always the preceding period.     * Useful for year-to-year comparisons.     * Formula: $\frac{\text{Current Period Value}}{\text{Preceding Period Value}} \times 100$     * Example 5.5 (Papillion Café):         * January Sales: $R14,980$; February Sales: $R16,433$         * February Link Relative: $\frac{16,433}{14,980} \times 100 = 109.7$

• Changing the Base Year: Necessary if the original base is too old, if comparing indexes with different bases, or if abnormal influences occurred during the original base period.     * Formula for Shifting Base:         $\text{New Index Number} = \frac{\text{Original (old) Index Number}}{\text{Old Index for the NEW base}} \times 100$     * Example 5.6 (Ford Motor Co.):         * Original base ($2002=100$): $2004$ index is $117.0$.         * To shift to $2004=100$: Divide all values by $117.0$         * $2000$ value: $\frac{74.2}{117.0} \times 100 = 63.4$