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Calculating a Weighted Price Index and Sustainable Government Debt
IB DP Economics — Unit 3: Macroeconomics | HL | 3.3 Macroeconomic Objectives
📋 What You Will Learn
- Calculate a weighted price index (CPI) using expenditure weights and price data
- Explain what the CPI measures and identify its limitations as a measure of inflation
- Define government debt and the budget balance, distinguishing between deficit and surplus
- Analyse the concept of sustainable government debt using the debt-to-GDP ratio
- Evaluate the conditions under which government debt is considered sustainable or unsustainable
- Apply quantitative skills to calculate debt-to-GDP ratios and interpret their macroeconomic significance
🔍 Guiding Question How do governments measure the cost of living and manage their finances — and at what point does borrowing become a threat to macroeconomic stability?
🌍 Why This Matters In 2023, the UK's Consumer Price Index hit 11.1% — the highest in 40 years — while the government's debt-to-GDP ratio exceeded 100% for the first time since the 1960s. Across the Eurozone, the European Central Bank raised interest rates to combat inflation while simultaneously worrying that higher borrowing costs would push heavily indebted governments like Italy and Greece toward fiscal crisis. These two issues — how we measure inflation and how much debt is too much — sit at the heart of modern macroeconomic policy.
Part 1 — The Consumer Price Index (CPI)
Inflation is not simply "prices going up." Different households buy different things, and some prices matter more to the average consumer than others. A 20% rise in the price of yachts affects very few people; a 20% rise in the price of bread affects everyone. The CPI accounts for this by weighting each good according to its importance in the typical household's spending.
📖 Definition — Consumer Price Index (CPI) The Consumer Price Index (CPI) is a weighted measure of the average change in prices of a representative basket of goods and services purchased by a typical household over time, expressed as an index relative to a base year.
📖 Definition — Inflation Rate The inflation rate is the percentage change in the CPI from one period to the next, measuring the rate at which the general price level is rising.
How the CPI Is Constructed
- Step 1 — Identify the basket: a representative sample of goods and services purchased by typical households is selected (e.g. food, housing, transport, clothing, healthcare)
- Step 2 — Assign weights: each category is given a weight reflecting its share of total household expenditure (e.g. housing = 30%, food = 20%, transport = 15%); weights sum to 100 (or 1.00)
- Step 3 — Collect price data: prices for each item in the basket are collected in the current period and compared to the base year
- Step 4 — Calculate price relatives: for each good, compute the price relative = (current price / base year price) × 100
- Step 5 — Apply weights: multiply each price relative by its weight, then sum across all categories to obtain the weighted price index
- Step 6 — Calculate inflation: inflation rate = ((CPI current − CPI previous) / CPI previous) × 100
Weighted Price Index Formula
\[ \text{CPI} = \sum \left( w_i \times \frac{P_{i,t}}{P_{i,0}} \times 100 \right) \] Where \( w_i \) is the weight of good \( i \) (as a decimal or percentage), \( P_{i,t} \) is the current price of good \( i \), and \( P_{i,0} \) is the base year price of good \( i \). The sum of all weights equals 1 (or 100 if expressed as percentages).
Inflation Rate Formula
\[ \text{Inflation Rate} = \frac{\text{CPI}_{\text{current}} - \text{CPI}_{\text{previous}}}{\text{CPI}_{\text{previous}}} \times 100 \]
📌 Base Year Convention In the base year, the CPI = 100 by definition. If the CPI rises to 112 in the following year, prices have risen by 12% on average since the base year. The base year is periodically updated to reflect changes in consumer spending patterns.
Limitations of the CPI as a Measure of Inflation
- Substitution bias: when prices rise,