Index numbers, the CPI and rates

The price of a bus ticket was $\text{€}\ 2.50$ in 2023 and $\text{€}\ 2.65$ 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 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 $\text{€}\ 11$ 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: $\text{€}\ 60$; 2023: $\text{€}\ 66$; 2024: $\text{€}\ 69$.

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 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?

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).

A female worker earned a net salary of $1,500$ € per month in 2018 and goes on to earn $1,720$ € per month in 2024. To assess whether she has gained or lost purchasing power, she consults the CSO's CPI series, but notices that in 2021 a base year change occurred:\n- Old series (base $2016 = 100$): Index in 2018 $= 103.5$; Index in 2021 $= 110.0$.\n- New series (base $2021 = 100$): Index in 2021 $= 100.0$; Index in 2024 $= 114.2$.\n\nTo calculate the cumulative inflation between 2018 and 2024, two approaches are proposed:\n- Method 1 (Linking to the old base): Multiply the indices of the new series by the linking coefficient $k = \frac{110.0}{100.0}$ to express the year 2024 in base 2016, and calculate the percentage change between 2018 and 2024.\n- Method 2 (Rebasing the series to base 2021): Change the base of the year 2018 directly to base 2021 by dividing its index by $110.0$ and multiplying by $100$, to then calculate the variation between 2018 and 2024 in that same base.\n\na) Fully develop both methods with the numerical data and demonstrate algebraically that they lead to exactly the same percentage of cumulative CPI variation.\nb) Compare both procedures and justify which of the two is preferable or more efficient if you had to update an entire company price table between 2015 and 2024.\nc) Using the calculated cumulative inflation, determine the real salary earned by the worker in 2024 at constant 2018 prices, and conclude with reasoning whether she has gained or lost purchasing power and by what exact percentage relative to her initial situation.

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 litre of milk cost $\text{€}\ 0.95$ 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 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.

In 2022, the rent for a flat was $\text{€}\ 850$ 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 euros». 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 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 table shows approximate data from the Labour 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.

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.