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Calculating a Weighted Price Index and Sustainable Government Debt
IB DP Economics — Unit 3: Macroeconomics | HL | 3.3 Macroeconomic Objectives [HL Extensions]
📋 What You Will Learn
- Calculate a weighted price index (CPI) using expenditure weights and price data
- Explain why weights are necessary for an accurate measure of inflation
- Interpret changes in a weighted price index and evaluate its limitations as a measure of inflation
- Define government debt and the budget balance, and distinguish between the deficit and the debt
- Analyse the concept of sustainable government debt using the debt-to-GDP ratio
- Evaluate the consequences of unsustainable debt and the policy options available to governments
🔍 Guiding Question How accurately can a single number capture the cost of living for an entire population — and at what point does a government's debt become a threat to macroeconomic stability?
🌍 Why This Matters In 2023, the UK's Consumer Price Index showed inflation at 6.7% — but a single parent spending 40% of their income on rent and food experienced a very different reality from a retiree whose largest expense was energy. Meanwhile, the UK's national debt crossed 100% of GDP for the first time since the 1960s, prompting serious debate about fiscal sustainability. Both issues — how we measure inflation and how we manage debt — sit at the heart of macroeconomic policy.
Part 1 — Calculating a Weighted Price Index
A simple average of price changes across goods would treat a 10% rise in the price of yachts the same as a 10% rise in the price of bread. This is clearly misleading — households spend far more on bread than yachts. A weighted price index corrects for this by assigning each good a weight proportional to its share of total household expenditure.
📖 Definition — Consumer Price Index (CPI) The Consumer Price Index (CPI) is a weighted price index that measures the average change in the price level of a representative basket of goods and services purchased by households over time, relative to a base year.
📖 Definition — Expenditure Weight An expenditure weight reflects the proportion of total household spending allocated to a particular category of goods or services. Weights are determined by household expenditure surveys and sum to 1 (or 1000 in some national systems).
Why Weights Matter
- Households spend different proportions of their income on different goods — a price index must reflect this to be meaningful
- If housing costs 30% of a typical household's budget, a 10% rise in rent contributes far more to inflation than a 10% rise in cinema tickets (0.5% of budget)
- Weights are updated periodically (usually annually) to reflect changing consumption patterns
- The base year is the reference point: the index is set to 100 in the base year, and subsequent values show the percentage change relative to that year
The Weighted Price Index Formula
Weighted Price Index Formula
\[ \text{WPI} = \sum \left( w_i \times \frac{P_{i,t}}{P_{i,0}} \times 100 \right) \] Where: \( w_i \) = expenditure weight for good \( i \) (weights sum to 1); \( P_{i,t} \) = price of good \( i \) in the current year; \( P_{i,0} \) = price of good \( i \) in the base year.
💡 Equivalent Formulation Some textbooks express this as: multiply the price relative (current price ÷ base price × 100) by the weight for each item, then sum all products. The result is the weighted price index for the current year. If the base year index = 100, then a current index of 108 means prices have risen by 8% on average since the base year.
Inflation Rate