Exercises: Index numbers, CPI and rates

The price of a bus ticket was $2,50\ \text{ €}$ in 2023 and $2,65\ \text{ €}$ 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 $11\ \text{ €}$ 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: $60\ \text{ €}$; 2023: $66\ \text{ €}$; 2024: $69\ \text{ €}$.

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.

In a regional socioeconomic report, the following data from the Labour Force Survey (EPA) are collected:\n- Population aged $16$ and over: $500.000$ people.\n- Employed people: $270.000$.\n- Unemployed people: $30.000$.\n\nMarcos, a high school student, writes in his paper: "The unemployment rate in the region is $6\,\%$, since it is calculated by dividing the $30.000$ unemployed people by the $500.000$ people of working age and multiplying by $100$".\n\na) Identify the conceptual error made by Marcos and write a brief explanation of why his approach does not fit the definition of the unemployment rate.\nb) Correctly calculate the activity rate and the unemployment rate for said region (expressed as a percentage rounded to the nearest hundredth).\nc) If next year $10.000$ of the people who were unemployed decide to stop actively seeking employment to enrol in full-time university studies, what will happen qualitatively to the unemployment rate (will it increase, decrease, or remain unchanged), assuming that the other variables stay constant? Justify your answer.

To analyse 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-on-year inflation rate (percentage change) has been for this family.

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 litre of milk cost $0,95\ \text{ €}$ 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 $1.650\ \text{ €}$ per month, what salary would they need to regain the purchasing power from three years prior?

In 2022, the rent for a flat was $850\ \text{ €}$ 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 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.

The statistics department of a city council prepares a local price index with three key components:\n- Food (weight $p_1 = 0,30$).\n- Housing and rent (weight $p_2 = 0,50$).\n- Municipal transport (weight $p_3 = 0,20$).\n\nAt the end of the year, the council announces that the municipality's overall CPI has risen by exactly $4,1\,\%$ compared to the base year (that is, the overall weighted index stands at $104,1$). Reviewing the sub-series, it is verified that the Housing and rent index rose by $3\,\%$ ($I_2 = 103$) and Transport became $2\,\%$ cheaper ($I_3 = 98$), but the record for the Food index ($I_1$) has been misplaced.\n\na) Set up a linear equation involving the weighted average of the CPI where the unknown is $I_1$.\nb) Solve the equation and calculate the index number for Food and its percentage rate of change.\nc) If the municipal government team had set a target for the municipality's overall inflation not to exceed $3,5\,\%$, what could have been, at most, the percentage increase in food prices while keeping the variations in housing and transport constant?

In an autonomous community, the labour market data for Year 1 according to the INE are as follows:\n- Population aged $16$ and over: $2.000.000$ people.\n- Activity rate: $60\,\%$.\n- Unemployment rate: $15\,\%$.\n\nIn Year 2, an economic recession occurs. The population aged $16$ and over remains at $2.000.000$ people, but two simultaneous phenomena take place: $20.000$ jobs are lost (the number of employed people decreases by $20.000$, initially moving into unemployment) and, at the same time, $40.000$ unemployed people succumb to the "discouraged worker effect" (they stop actively looking for work and leave the active population, becoming inactive).\n\na) Determine for Year 1 the active population, the number of employed people, and the number of unemployed people.\nb) Determine for Year 2 the new active population, the new number of employed people, and the new number of unemployed people. From these, find the new activity rate and the new unemployment rate (rounding to two decimal places).\nc) Interpret the result: why has the unemployment rate decreased even though net jobs have been lost? Explain the risk of assessing the health of employment solely by looking at the unemployment rate without considering the activity rate.

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, $100\ \text{ €}$) 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 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.

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 INE'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.

A real estate platform analyses the series of rental prices per square metre in a capital city between 2019 and 2024, taking $2019 = 100$ as the base year:\n- In 2020, the price experienced a decrease of $4\,\%$ relative to 2019.\n- In 2021, the price rose by $5\,\%$ relative to 2020.\n- During the 2021-2023 biennium, the price increased at a constant average annual rate of change (AARC) of $6\,\%$ each year.\n- Looking ahead to 2024, the company aims to adjust its rates so that the average annual rate of change over the entire five-year period (the full 2019-2024 period, that is, in $5$ years) is exactly $4\,\%$ per year.\n\na) Model the evolution of the series by calculating the price index number in 2020, 2021, and 2023 with base 2019 (round the indices to two decimal places).\nb) Write the equation that allows obtaining the price index corresponding to the year 2024 to satisfy the condition of a $4\,\%$ five-year AARC, and calculate the value of said index $I_{2024}$.\nc) From the previous results, deduce what exact percentage change must be applied to the rent in 2024 compared to 2023 prices.\nd) A sales consultant proposes: "Since in 2020 it changed by $-4\,\%$, in 2021 by $+5\,\%$, in 2022 by $+6\,\%$, and in 2023 by $+6\,\%$, the cumulative sum is $-4 + 5 + 6 + 6 = 13\,\%$. To achieve an average of $4\,\%$ per year over five years (which adds up to $20\,\%$), in 2024 it will be enough to increase it by $7\,\%$ (since $13 + 7 = 20$)". Argue with mathematical rigour why the consultant's reasoning is fallacious in financial mathematics and index numbers, and indicate whether their estimate falls above or below the actual required value.