Exercises: Index numbers, CPI and rates

The price of a bus ticket was $\$50$ in 2023 and $\$53$ 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 set 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 set menu cost $\$220$ 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?

In a country of $2.1$ million inhabitants, the birth rate is $7.1$ births per $1,000$ inhabitants.

1. a) Approximately how many births were recorded that year?

1. b) If the death rate was $9.3$ per $1,000$ inhabitants, is the population growing or shrinking due to natural causes? Quantify it.

The series of index numbers for the price of urban transport, with base year 2018, is: 2018: $100$; 2019: $103.2$; 2020: $107.5$; 2021: $112.1$.

1. a) Construct the same series using 2021 as the base year (index $100$ in 2021), rounding to one decimal place.

1. b) What information is preserved and what changes when changing the base year?

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 $\$19$ 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 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?

In 2022, the rent for an apartment was $\$17\,000$ per month. The contract is reviewed 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 pesos". 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 municipality of $48,300$ inhabitants, $415$ births and $502$ deaths were recorded in a year.

1. a) Calculate the birth rate and the death rate (per $1,000$ inhabitants), rounded to two decimal places.

1. b) Calculate the natural increase (births minus deaths) and express it also as a rate per $1,000$ inhabitants. Is the population growing or shrinking due to natural causes?

1. c) If the total population, however, increased by $600$ people that year, what other demographic component explains the difference? Quantify it.

The figure shows the evolution of a price index with base $100$ in 2021.

1. a) What was the total percentage change in prices between 2021 and 2025?

1. b) In which year did the largest annual rise occur? Calculate it from the indices; do not estimate it by sight.

1. c) Calculate the average annual rate of change for the period (the rate which, if repeated over the four years, would give the same final index). Round to two decimal places.

A supermarket chain has a series of index numbers for its sales with base year 2020 and another with base year 2023, but is missing the link between them. The known data are:

• Base 2020: 2021 index $= 104$, 2022 index $= 109$, 2023 index $= 115$.

• Base 2023: 2024 index $= 106$, 2025 index $= 110$.

1. a) Calculate the 2024 and 2025 indices with base 2020 (that is, link the two series).

1. b) What was the total percentage change in sales between 2020 and 2025?

1. c) And what was the average annual change in that period? Express it as a percentage rounded to one decimal place (use the root of the overall index).

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, $\$2\,000$) 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.

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 drink: weighting $25 \%$, rise of $9 \%$.

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

• Transport: 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.

The table shows approximate data from the Labor Force Survey for two regions in a quarter:

• Region A: active population $2,400,000$ people, unemployed $312,000$, employed $2,088,000$.

• Region B: active population $950,000$ people, unemployed $123,500$, employed $826,500$.

1. a) Calculate the unemployment rate for each region, rounded to two decimal places.

1. b) In region A, the population aged $16$ or over is $4,300,000$ people. Calculate its participation rate, rounded to two decimal places.

1. c) A headline reads: «Region B has less unemployment because it has fewer unemployed people». Critique the headline using the calculated rates and explain why rates allow us to compare populations of different sizes.