Index numbers, the CPI and rates

The price of a bus ticket was $\$2.50$ in 2023 and $\$2.65$ in 2024.

1. a) Calculate the price index number in 2024 with base year 2023.

1. b) What was the percentage change in price?

The index number for the prix fixe lunch menu price in a city, with base year 2021, is $108.5$ in 2024.

1. a) Explain in your own words what this value means.

1. b) If a prix fixe menu cost $\$11$ in 2021, how much does it cost in 2024 if its price has followed this index? Round to the nearest cent.

The table shows the price of a video game across three years: 2022: $\$60$; 2023: $\$66$; 2024: $\$69$.

1. a) Construct the series of index numbers with base year 2022.

1. b) In which year did the price increase the most, as a percentage, compared with the previous year?

The figure compares the evolution of the CPI and the wage index of a country (both with base $100$ three years prior).

1. a) What was the total percentage rise in prices and in wages over the period?

1. b) Have wages gained or lost purchasing power? Calculate the real index of variation (wage index divided by the CPI, multiplied by $100$) for the last year and interpret it.

1. c) If in the last year a worker earned $\$1,650$ per month, what salary would they need to regain the purchasing power from three years prior?

To analyze the shopping basket of an average family, only three consumption groups are considered with their corresponding weightings (weights) and price indices relative to the previous year:\n- Food: weighting $p_1 = 0.35$; price index $I_1 = 108$ ($8\,\%$ increase).\n- Housing and utilities: weighting $p_2 = 0.45$; price index $I_2 = 102$ ($2\,\%$ increase).\n- Leisure and transport: weighting $p_3 = 0.20$; price index $I_3 = 99$ ($1\,\%$ decrease).\n\na) Before calculating the exact numerical result, make a reasoned estimate of the interval in which the overall CPI will fall. Will it be closer to $102$, $105$, or $108$? Justify your estimate based on the weight of each expenditure group.\nb) Calculate the exact weighted CPI index.\nc) Check whether your initial estimate was consistent and deduce what the overall year-over-year inflation rate (percentage change) has been for this family.

The annual CPI for three consecutive years was $3.5 \%$, $2.8 \%$ and $4.1 \%$.

1. a) Calculate the chained index of variation for the entire three-year period and the total cumulative rise, rounded to two decimal places.

1. b) If a liter of milk cost $\$0.95$ at the start of the first year and its price has increased by exactly the CPI, how much does it cost at the end of the third year? Round to the nearest cent.

The CPI of a country is calculated as a weighted average of the indices of several expenditure groups. In a given year, the data are:

• Food and beverages: weighting $25 \%$, rise of $9 \%$.

• Housing and utilities: weighting $30 \%$, rise of $4 \%$.

• Transportation: weighting $15 \%$, rise of $6 \%$.

• Other groups: weighting $30 \%$, rise of $2 \%$.

1. a) Calculate the overall CPI for the year, rounded to two decimal places.

1. b) A worker whose salary rose by $3.5 \%$ that year claims that they have "lost purchasing power." Are they right? Justify your answer by calculating the real index of variation of their salary.

1. c) Another worker spends half of their salary on housing and the other half equally among the remaining items. Explain why their "personal inflation" may be different from the official CPI.

In 2022, the rent for an apartment was $\$850$ per month. The lease is adjusted every year with the CPI: in 2023 the CPI was $3.1 \%$ and in 2024, $2.6 \%$.

1. a) Calculate the monthly rent in 2024 after both adjustments, rounded to the nearest cent.

1. b) The tenant wants to compare the 2024 rent «in 2022 dollars». Since prices have increased at the same rate as their rent, explain which index they would use and what conclusion they reach.

1. c) If in 2025 the CPI is $5 \%$ and the landlord decides not to increase the rent, by what percentage has the rent fallen in real terms? Round to two decimal places.

In a televised debate, a pundit states: «the CPI rose by $5 \%$ in 2023 and fell by $5 \%$ in 2024, so prices have returned exactly to where they were in 2022».

1. a) Show using a specific price (for example, $\$100$) that the statement is false, and calculate the overall index of variation for the two-year period.

1. b) By what percentage should prices have fallen in 2024 to return exactly to the 2022 level? Round to two decimal places.

1. c) Explain, in a sentence that any citizen would understand, why a rise of $5 \%$ and a fall of $5 \%$ do not cancel each other out.