Introduction and Growth Rates
Statistics I
Course Introduction
Core Information
Instructor: Paulo Fagandini
:email: pfagandini@iscal.ipl.pt
Main source for course material: Moodle@ISCAL
Official communication channel: :email: Institutional Email
Bibliography
Topic 1:
:book: Gancho Custódio, S. et al. (2022) Números Índices, Edições Sílabo
Topic 2:
:book: Murteira, B.; Silva Ribeiro, C.; Andrade e Silva, J. & Pimenta, C., Introdução à Estatística, Escolar Editora, McGraw-Hill, 2010
Topic 3:
:book: Ferreira, T., Custódio, S.G. (2023) Modelos Probabilísticos, Edições Sílabo
Bibliography (EN)
Topic 1:
:book: The making of index numbers. 1st Ed 1922. Irving Fisher.
:book: A Practical Introduction to Index Numbers. 1st Ed 2015. Jeff Ralph, Rob O’Neill, Joe Winton.
Topics 2 and 3:
:book: Statistics for Business and Economics Global Edition. 10th Ed 2022. Paul Newbold, William Carlson, and Betty Thorne.
This last book is a bit expensive, but it can be rented for less money here.
Continuous Assessment
| Assessment Element | Weight (%) | Duration | Syllabus | Date |
|---|---|---|---|---|
| Midterm 1 | 60% | 80 minutes | Topics 1 and 2 | Nov 5th |
| Midterm 2 | 40% | 80 minutes | Topic 3 | Dec 10th |
Each test has a minimum grade of 7.00 (out of 20), not 6.9, not 6.95.
Comprehensive Exam
Students may opt, on the day of the Exam, to take an Exam worth 100% of the grade. This Exam will cover Topics 1, 2, and 3.
If you score less than 7.00 in the Midterm, or if you cannot attend the midterm for whatever reason, you will have to do the Comprehensive Exam.
Disclaimer
These slides are a free translation and adaptation from the slide deck for Estatística I by Prof. Sandra Custódio and Prof. Teresa Ferreira from the Lisbon Accounting and Business School, Polytechnic University of Lisbon.
Growth Rates
Example: Apple Stock
Source: Yahoo Finance
| Date | Close |
|---|---|
| 2020 | 132.69 |
| 2021 | 177.57 |
| 2022 | 129.93 |
| 2023 | 192.53 |
| 2024 | 250.42 |
| 2025 | 271.86 |
| 2026 | 333.02 |
Example: Apple Stock
Let the close price for Apple be represented by the variable \(y_t\), so \(y_{2020}\) is the close price we got for 2020 (i.e. 132.69).
How much did \(y\) grow between 2020 and 2021?
\[\Delta y_{2021} = y_{2021} - y_{2020} = 177.57 - 132.69\] \[=44.88\]
Or we could say
\[y_{2021} = y_{2020} + \Delta y_{2021}\] \[177.57 = 132.69 + 44.88\]
Example: Apple Stock
\[y_{2021}=y_{2020}+\Delta y_{2021}\]
\[y_{2021}=y_{2020}\left(1+\frac{\Delta y_{2021}}{y_{2020}}\right)\]
\[y_{2021}=y_{2020}\left(1+\delta^y_{2021}\right)\]
This, \(\delta_{2021}^y\), is the growth rate of \(y\) at year 2021.
Growth Rate
Growth Rate
You can write the growth rate as a decimal or as a percentage. If you use decimal notation, consider at least 4 places, if you use percentage you use at least 2. Example: 0.0123 or 1.23%.
Interpretation: What is the percentage change for the variable \(y\) between \(t-1\) and \(t\).
Example
\[177.57 = 132.69+(177.57 - 132.69)\]
\[177.57 =132.69\left(1+\frac{44.88}{132.69}\right)\]
\[177.57=132.69\left(1+0.3382\right)\]
In this case \(\delta_{2021}^y =0.3382 = 33.82\%\).
Interpretation: Between 2020 and 2021, the price for Apple increased 33.82%.
Growth Rates
More generally \[\delta_{t+k|t}=\frac{y_{t+k}-y_t}{y_t}\]
Or \[y_{t+k}=y_t(1+\delta_{t+k|t})\]
(homework: Show step by step how you go from one to the other.)
Note: In this case \(\delta_{t+k|t}\) is what percentage the variable changed from \(t\) to \(t+k\), and \(k\) is the number of periods.
Example
We had \(y_{2020}=132.69\), and \(y_{2024} =250.42\). We will try to find \(\delta_{2024|2020}^y\)
- What is the value of \(k\) in this case?
\[k=2024-2020=4\]
- What is the growth rate between \(2020\) and \(2024\)?
\[\delta_{2024|2020}^y = \frac{250.42 - 132.69}{132.69} = 0.8873 = 88.73\%\]
For Apple, the stock price grew 88.73% between 2020 and 2024, or the stock price for Apple in 2024 is 88.73% larger than in 2020.
Growth Rates
Warning
You should be careful with your interpretation and use of this cumulative growth rate, because it does not mean that \(y\) grew \(\delta_{t+k|t}^y\) per period between \(t\) and \(t+k\).
Growth Rates
Growth Rates
Let’s start with \(y_t\)… and the traditional growth rates \(\delta_t\):
\[y_{t+1} = y_t(1+\delta_{t+1})\]
\[y_{t+2} = y_{t+1}(1+\delta_{t+2})\]
\[y_{t+3} = y_{t+2}(1+\delta_{t+3})\]
Can we write \(y_{t+2}\) as a function of \(y_t\)?
\[y_{t+2} = y_{t+1}(1+\delta_{t+2}) = y_{t}(1+\delta_{t+1})(1+\delta_{t+2})\]
And \(y_{t+3}\)?
\[y_{t+3} = y_{t}(1+\delta_{t+1})(1+\delta_{t+2})(1+\delta_{t+3})\]
Growth Rates
Generalizing
\[y_{t+k}=y_t (1+\delta_{t+1})...(1+\delta_{t+k-1})(1+\delta_{t+k})\]
When computing the average growth rate, we are trying to find \(\overline{\delta}\) such that if the growth rate was the same every period, it would have taken \(y_t\) to the value of \(y_{t+k}\) all the same:
\[y_{t+k}=y_t(1+\overline{\delta})(1+\overline{\delta})...(1+\overline{\delta})\]
How many times is \((1+\overline{\delta})\) multiplied in the expression?
Growth Rates
\[y_{t+k} = y_t \left(1+\overline{\delta}\right)^k\]
\[\frac{y_{t+k}}{y_t} = \left(1+\overline{\delta}\right)^k\]
\[\left(\frac{y_{t+k}}{y_t}\right)^{1/k} = \left(1+\overline{\delta}\right)\]
\[\left(\frac{y_{t+k}}{y_t}\right)^{1/k} - 1 = \overline{\delta}\]
\[\overline{\delta}_{t+k|t}=\left(\frac{y_{t+k}}{y_t}\right)^{1/k} - 1\]
Example
We had \[\delta_{2024|2020}^y = \frac{250.42 - 132.69}{132.69} = 0.8873 = 88.73\%\]
But what now is the average growth rate between 2020 and 2024?
Remember \(k = 4\)
\[ \overline{\delta} = \left(\frac{250.42}{132.69}\right)^{1/4} - 1 \approx 0.1721= 17.21\% \]
Interpretation: On average, between 2020 and 2023 \(y\) grew 17.21% every year.
Example
Note that \[ (1+17.21\%)^4 = (1+0.1721)^4 \approx 1.8873\approx 1 + 88.73\% \]
❓ Growth Rates · Question 1
A variable grows 10% in one year and then falls 10% in the next. Over the two years together it:
A. fell 1%
B. rose 1%
C. is unchanged
D. fell 10%
✅ A. Growth rates compound, they do not add: \((1+0.1)(1-0.1)=0.99\), so the variable ends 1% below where it started.
❓ Growth Rates · Question 2
A variable goes from 100 to 150 over 5 years. Its average growth rate is closest to:
A. 10%
B. 8.45%
C. 50%
D. 5%
✅ B. \(\overline{\delta}=\left(\frac{150}{100}\right)^{1/5}-1=1.5^{0.2}-1\approx 0.0845\), that is 8.45% per year.
✏️ Growth Rates · Question 3
A firm sold 240 units in 2020 and 300 units in 2023.
Compute:
(i) the cumulative growth rate over the three years;
(ii) the average annual growth rate.
✅ Growth Rates · Solution
(i) \(\delta_{2023|2020}=\frac{300-240}{240}=0.25\), a cumulative 25%.
(ii) Here \(k=3\), so \(\overline{\delta}=\left(\frac{300}{240}\right)^{1/3}-1=1.25^{1/3}-1\approx 0.0772\), about 7.72% per year.
Note 7.72% per year for 3 years is not 25/3, precisely because growth compounds.
Index Numbers: Introduction and Simple Index Numbers
Motivation
Which stock would you have purchased in 2016-10-03?
Motivation
What’s the important question here?
Which one grew more! For this, levels are not as relevant as their evolution over time.
For example, say we have prices for two stocks A and B. Say we want to hold these stocks for only one period, which one would choose to invest your hard earned :dollar:?
How many stocks will you be able to buy of each stock, if their prices are \(a_t\) and \(b_t\), and you have US$1000?
\(n_a=\frac{1000}{a_t}\) of stock A and \(n_b=\frac{1000}{b_t}\) of stock \(B\). Note that \(n_a\) and \(n_b\) are the amount of stock you can buy of each.
Motivation
Now is a new day :sunrise_over_mountains:! Prices are now \(a_{t+1}\) and \(b_{t+1}\), how much is your portfolio worth today?
If you bought stock A: \(a_{t+1} n_a = a_{t+1}\frac{1000}{a_t}\)
If you bought stock B: \(b_{t+1} n_b = b_{t+1}\frac{1000}{b_t}\)
Note that the 1000 is fixed, so it is just what we invested, it is not going to change anything in our decision on buying A vs B, what is really important is \(\frac{a_{t+1}}{a_t}\) and \(\frac{b_{t+1}}{b_t}\), that is the growth rate of prices for each stock, as
\[\frac{a_{t+1}}{a_t}=1+\delta_{t+1}^a \quad\text{and}\quad \frac{b_{t+1}}{b_t}=1+\delta_{t+1}^b\]
Motivation
You did not sell and…
Now is (another) new day :sunrise_over_mountains:! Prices are now \(a_{t+2}\) and \(b_{t+2}\), how much is your portfolio worth today?
If you bought stock A: \(a_{t+2} n_a = a_{t+2}\frac{1000}{a_t}\)
If you bought stock B: \(b_{t+2} n_b = b_{t+2}\frac{1000}{b_t}\)
blah, blah…
\[\frac{a_{t+2}}{a_t}=1+\delta_{t+2}^a \quad\text{and}\quad \frac{b_{t+2}}{b_t}=1+\delta_{t+2}^b\]
If we wait two days, we only care how is the price today compared to when we made the purchase. We could make instead a plot of \(1+\delta_{t+k}\) to see which grew more!
Motivation
Now we have a much clear picture. We can compare their evolution starting in 2016-10-03.
What can you read from the plot?
Motivation
Note that for 2016-10-03, we would have, for A and B:
\[1+\frac{a_t-a_t}{a_t}=1\quad\text{and}\quad 1+\frac{b_t-b_t}{b_t}=1\]
We just built and index number!
By convention though, instead of using 1 for the reference date, we will use 100 (as in 100%) for the reference day (we will put a name to this in a few moments).
When our new series was 2, we understood that \(1+\delta_{t+k}=2\) or \(\delta_{t+k}=1\), i.e. the price grew 100%, or doubled! Now, with the new notation, we would have that the new series would take the value of 200, i.e. the price is now 200% the value it had at our initial date.
Index Numbers
In our previous example, the base day would have been 2016-10-03. Note that in the base period \[I_0=\frac{x_0}{x_0}\times 100 = 100\]
Index Numbers
Index Numbers are defined by their underlying variable. This can change over time, or it can change according to another dimension, like geography, or both! We will see examples.
Also, an Index Number can have a fixed base or a moving base.
Index Numbers
WarningIndex Numbers
- Simple and Composite are mutually exclusive
- Fixed and Moving are mutually exclusive
- Rolling always needs a chronological dimension
Examples
| Type | Example |
|---|---|
| Simple | Single stock price |
| Composite | SP500 (market index), CPI |
| Fixed Base | GDP, CPI, market index |
| Rolling Base | Some measures of GDP, Commodities Futures |
| Chronological | Stock price over time, CPI, GDP |
| Geographical | BigMac Index |
Index Numbers
Importantly, an index number is no bueno to reflect the level of a variable, but its evolution or relative value against a base.
Rolling Base
Index Numbers
Index numbers allow us to easily compute the growth rate between any two periods, just like we would do with the underlying variable.
1. Simple Index Number
\[1+\delta_{t+k,t}=\frac{x_{t+k}}{x_t}=\frac{\frac{x_{t+k}}{x_0}}{\frac{x_t}{x_0}}=\frac{I_{t+k}}{I_t}\]
Index Numbers
2. Chain Index Number
\[1+\delta_{t+k,t}=\frac{x_{t+k}}{x_t}=\frac{x_{t+k}}{x_{t+k-1}}\,\frac{x_{t+k-1}}{x_{t+k-2}}\cdots\frac{x_{t+1}}{x_{t}}\]
\[1+\delta_{t+k,t}=I_{t+k}\times I_{t+k-1}\times \cdots \times I_{t+1}\]
Index Numbers
WarningDo not use 100 index when multiplying chain or link indices, use the decimal notation.
Note that if the underlying variable does not change at all, you should expect the same value, \(x_t=x_{t+1}=x_{t+2}\)
In this case \(I_{t+1}=\frac{x_{t+1}}{x_t}=1\) and \(I_{t+2}=\frac{x_{t+2}}{x_{t+1}}=1\), and therefore \[1+\delta_{t+2,t}=1\times 1= 1\]
This is the correct way to do it :white_check_mark:.
Index Numbers
WarningAnd if we multiplied the indices with the 100?
If instead you used the 100 to multiply: \[1+\delta_{t+2,t}=100\times 100= 10000\] and we get that \(x\) grew 100 times! (:x: because we know that \(x_t=x_{t+1}=x_{t+2}\))
Index Numbers
We can always transform a chain index number into a fixed base index number.
Chose a reference, for example for period 3.
Remember: what you want to achieve: \(\forall t, I'_t=\frac{x_t}{x_3}\)
Remember: that we have \(I_t=\frac{x_t}{x_{t-1}}\)
Use what we did just before:
\[ I_t' = I_t I_{t-1} ...I_{4} = \frac{x_t}{x_{t-1}}\frac{x_{t-1}}{x_{t-2}}...\frac{x_4}{x_3}=\frac{x_t}{x_3}=I'_t\]
Example
Suppose we want to convert this Chain Index into Fix Base index, with the base year in 2021. This will be our 100.
| 2021 | 2022 | 2023 | 2024 | 2025 | 2026 | |
|---|---|---|---|---|---|---|
| Price | 30 | 34 | 37 | 35 | 36 | 39 |
| \(\frac{34}{30} 100\) | \(\frac{37}{34} 100\) | \(\frac{35}{37} 100\) | \(\frac{36}{35} 100\) | \(\frac{39}{36} 100\) | ||
| \(I_{chain}\) | 113.3 | 108.8 | 94.6 | 97.1 | 108.3 | |
| \(100\frac{113.3}{100}\) | \(113.3\frac{108.8}{100}\) | \(123.3\frac{94.6}{100}\) | \(116.7 \frac{97.1}{100}\) | \(120\frac{108.3}{100}\) | ||
| \(I_{t|2021}\) | 100 | 113.3 | 123.3 | 116.7 | 120 | 130 |
Example
Note that, for example in 2025, we have \(I_{2025|2021}=116.7\), which was computed as \(123.3\times\frac{94.6}{100}\). But \(123.3\) was the base 2021 index of the previous period, which in turn was computed as \(113.3\frac{108.8}{100}\), so in reality:
\[116.7=123.3\frac{94.6}{100}=113.3\frac{108.8}{100}\frac{94.6}{100}=100\frac{113.3}{100}\frac{108.8}{100}\frac{94.6}{100}\]
If you pay attention, you will see that we have there the product of the chain Index in 2022, 2023, and 2024. We divided by 100 because, to multiply, we need to divide by 100 before.
Base Change
In case we have the data, we just need to compute the Index with the base that we want, however, if we do not have the data, and we just have a Fix Base index, we can do it all the same!
Say we have an index with base 0. Maybe we want to change it to base 2.
Base Change
| period | Index | Formula |
|---|---|---|
| 0 | \(I_{0|0}\) | \(\frac{x_0}{x_0}\) |
| 1 | \(I_{1|0}\) | \(\frac{x_1}{x_0}\) |
| 2 | \(I_{2|0}\) | \(\frac{x_2}{x_0}\) |
| 3 | \(I_{3|0}\) | \(\frac{x_3}{x_0}\) |
| … | … | … |
| t | \(I_{t|0}\) | \(\frac{x_t}{x_0}\) |
| period | Index | Formula |
|---|---|---|
| 0 | \(I_{0|2}\) | \(\frac{x_0}{x_2}\) |
| 1 | \(I_{1|2}\) | \(\frac{x_1}{x_2}\) |
| 2 | \(I_{2|2}\) | \(\frac{x_2}{x_2}\) |
| 3 | \(I_{3|2}\) | \(\frac{x_3}{x_2}\) |
| … | … | … |
| t | \(I_{t|2}\) | \(\frac{x_t}{x_2}\) |
Base Change
But how can we transform them, if we do not know \(x_i\) \(\forall i\)?
\[I_{t|2}=\frac{x_t}{x_2}=1\frac{x_t}{x_2}=\frac{1/x_0}{1/x_0}\frac{x_t}{x_2}=\frac{\frac{x_t}{x_0}}{\frac{x_2}{x_0}}=\frac{I_{t|0}}{I_{2|0}}\]
And that is how we can re-base an index without knowing the value for the underlying variable (\(x_t\))!
TipMixing up stuff
Previously we covered how to go from a chain index to a fixed base one. Now we saw how to change the base of a fixed base index. With both things together you can turn a chain index into fixed base one, with base you want!
Example
| 2021 | 2022 | 2023 | 2024 | 2025 | 2026 | |
|---|---|---|---|---|---|---|
| Price | 30 | 34 | 37 | 35 | 36 | 39 |
| \(I_{t|2021}\) | 100 | 113.3 | 123.3 | 116.7 | 120 | 130 |
| \(\frac{100}{123.3}\) | \(\frac{113.3}{123.3}\) | \(\frac{123.3}{123.3}\) | \(\frac{116.7}{123.3}\) | \(\frac{120.0}{123.3}\) | \(\frac{130.0}{123.3}\) | |
| \(I_{t|2023}\) | 81.1 | 91.9 | 100 | 94.6 | 97.3 | 105.4 |
New information? Yes! For example, without any computation we know that the price in 2026 was 30% larger than in 2021, and 5.4% larger than in 2023
Does changing the base change the story?
Same prices as above. Move the slider to pick which year is the base, and watch what moves and what does not.
Base year:
What the chart showed
Changing the base slides the whole curve up or down: the orange 100 line stays where it is, and a different year gets pinned to it.
But the shape is untouched, and every growth rate in the panel stays exactly the same. Re-basing changes the yardstick, never the underlying evolution.
If two countries publish the same price series on different bases, they are not disagreeing about inflation. Divide one index by another of the same series and the base cancels, which is precisely the formula we just derived.
Tests for good Simple Index Numbers
We can test the properties for Simple Index Numbers. These represent desirable properties for good index numbers.
- Identity test: If the variable in the current period is the same as in the base period, the index should be 1 (or 100). Of course this implies \[I_{0|0}=1\].
- Proportionality test: If the variable increases by \(k\), the index should increase by \(k\). \[x_t=kx_0\ \Rightarrow\ i_{t|0}=k\]
Tests for good Simple Index Numbers
- Time Reversal test: If you exchange the base between two periods, then the index in one base, should be the inverse of the other: \[i_{t|0}=\frac{1}{i{0|t}}\ \Leftrightarrow\ i_{t|0}i_{0|t}=1\]
- Factor Reversal test: If we have two index numbers, one for variable \(x\) and another for \(y\), then if \(z=x\times y\) we should have \[i_{t|0}^z=i_{t|0}^x\times i_{t|0}^y\]
Tests for good Simple Index Numbers
Circularity test: The index at \(t\) base \(0\) should be the product of all the link index numbers up to \(t\), \[i_{t|0}=i_{t|t-1}\times i_{t-1|t-2}...i_{2|1}\times i_{1|0}\] This formalizes the relationship between fixed base and chain index numbers.
Homogeneity test: If the variable, for every \(t\) is multiplied by a constant, the index should not be affected.
Tests for good Simple Index Numbers
- Well-definedness: The index number must be meaningful, and as prices and quantities usually are larger than zero, index numbers must be positive. Even further, a good index number should not take the value of \(\infty\).
Ask yourself, what information are you getting here?
A value of \(0\) might also be problematic, although it is not totally meaningless. The problem arises when you want to deal with chain index numbers, where a \(0\) would indeterminate the rest of the chain. Also, you could never use as a base period, a period where the variable took the value of \(0\).
❓ Simple Index Numbers · Question 1
A simple price index with base 2020 reads 125 in 2024. This tells you that the price:
A. rose 125%
B. rose 25%
C. is 125 euros
D. nothing, without the base price
✅ B. The index is \(\frac{x_t}{x_0}\times 100\). A reading of 125 means \(\frac{x_t}{x_0}=1.25\), so the price is 25% above the base.
❓ Simple Index Numbers · Question 2
To turn a chain (link) index into a fixed base index you:
A. multiply them, in decimal notation
B. add the link indices
C. average them
D. subtract 100 from each
✅ A. Link indices multiply, and you must divide by 100 first, otherwise each extra factor of 100 inflates the answer.
✏️ Simple Index Numbers · Question 3
The price of a good was 30 in year 1, 34 in year 2 and 37 in year 3.
Compute the fixed base index with base year 1 for all three years, and the link index for year 3.
✅ Simple Index Numbers · Solution
Fixed base year 1: \(I_{1|1}=100\), \(I_{2|1}=\frac{34}{30}\times 100=113.3\), \(I_{3|1}=\frac{37}{30}\times 100=123.3\).
Link index for year 3: \(i_{3|2}=\frac{37}{34}\times 100=108.8\).
Check: \(100\times\frac{113.3}{100}\times\frac{108.8}{100}\approx 123.3\) ✅ (chain the link indices in decimal, never on the 100 scale)
Index Numbers: Composite Index Numbers
Composite Index Numbers
Composite Index Numbers
This is a great opportunity to introduce another classification for our index numbers:
Composite Index Numbers
Typically, values are collected in nominal terms, i.e. current prices, which is a value.
\[v_t=p_t\times q_t\]
Note that, if \(\delta_{t+1}^v\) is the growth rate of \(v_t\):
\[v_{t+1} = v_t\left(1+\delta_{t+1}^v\right)\]
and obviously, it is also true that:
\[v_{t+1} = p_{t+1}\times q_{t+1}\]
Composite Index Numbers
But if we let \(\delta_{t+1}^p\) be the growth rate of prices, and \(\delta_{t+1}^p\) the growth rate of quantities we can rewrite \(v_{t+1}\) as:
\[v_{t+1}=p_t\left(1+\delta_{t+1}^p\right)q_t\left(1+\delta_{t+1}^q\right)\]
\[v_{t+1} = p_tq_t(1+\delta_{t+1}^p)(1+\delta_{t+1}^q)\]
\[v_{t+1} = v_t(1+\delta_{t+1}^p)(1+\delta_{t+1}^q)\]
\[v_{t}(1+\delta_{t+1}^v) = v_t(1+\delta_{t+1}^p)(1+\delta_{t+1}^q)\]
:bulb:
Composite Index Numbers
\[(1+\delta_{t+1}^v) = (1+\delta_{t+1}^p)(1+\delta_{t+1}^q)\]
\(\delta_t^v\) is what we call nominal change rate, while \(\delta_t^q\) is what we call real change rate. We can find out the real change rate (\(\delta^q\)), if we know the nominal change rate (\(\delta^v\)), and the prices change rate (\(\delta^p\)):
\[\left(1+\delta_{t}^q\right) = \frac{\left(1+\delta_t^v\right)}{\left(1+\delta_t^p\right)}\] \[\delta_{t}^q = \frac{\left(1+\delta_t^v\right)}{\left(1+\delta_t^p\right)}-1\]
Composite Index Numbers
Suppose we have index numbers for \(v\), \(q\), and \(p\), all with base \(b\) (remember you can re-base your index numbers easily if they do not share the same base).
\[I_{t|b}^q = \frac{I_{t|b}^v}{I_{t|b}^p}\]
\[\frac{q_t}{q_b} = \frac{\frac{v_t}{v_b}}{\frac{p_t}{p_b}}\] \[\frac{q_t}{q_b} = \frac{\frac{p_tq_t}{p_bq_b}}{\frac{p_t}{p_b}}=\frac{p_tq_t}{p_bq_b}\frac{p_b}{p_t}=\frac{q_t}{q_b}\]
Example
| 2022 | 2023 | 2024 | 2025 | 2026 | |
|---|---|---|---|---|---|
| Sales Index | 99 | 100 | 107 | 110 | 111 |
| Price Index | 95 | 100 | 102 | 106 | 109 |
Let’s find the quantities index, with base 2023. Immediately we can fix \(i_{t|b}^q=100\).
\[I_{2022}^q=\frac{I_{2022}^v}{I_{2022}^p}=\frac{99}{95}=1.04\]
Multiplying by 100, \(I^q_{2022} = 104\), so quantities decreased from 2022 to 2023!
Example
| 2022 | 2023 | 2024 | 2025 | 2026 | |
|---|---|---|---|---|---|
| Quantities Index | 104.21 | 100 | 104.9 | 103.77 | 101.83 |
Now we can compute what was the real change rate between 2025 and 2026:
\[\delta_{2026}=\frac{102}{104}-1=-0.0187=-1.87\%\]
Real sales decreased 1.87% from 2025 to 2026.
Example
Let’s check:
\[\delta_{2026}^q=\frac{(1+\delta_{2026}^v)}{(1+\delta_{2026}^p)}-1=\frac{111/110}{109/106}-1=0.9813-1\] \[\delta_{2026}^q=-0.0187=-1.87\%\]
Baskets
A very important question: What are the weights that the different goods should have in the basket?
Aggregate Index Numbers
From the family of formulas to create weighted composite index numbers, we will focus on the two most common:
Laspeyres: uses base-period quantities (prices) as weights.
Paasche: uses current-period quantities (prices) as weights.
These are used to measure the evolution of prices (quantities).
Kids’ Explanation
Old basket (🧺\(_{past}\)):
- 🍎 10 apples
- 🍊 5 oranges
Old prices (🏷️\(_{past}\)):
- \(p_🍎=1\)
- \(p_🍊=0.8\)
New basket (🧺\(_{today}\)):
- 🍎 8 apples
- 🍊 6 oranges
New prices (🏷️\(_{today}\)):
- \(p_🍎=1.2\)
- \(p_🍊=1.1\)
Let’s consider that the base year is past.
Kids’ Explanation - Prices
Laspeyres - Price:
Use always \(🧺_{past}\), but relevant :label:! Laspeyres Price Index (LPI):
\[🧺_{past}|\text{🏷️}_{past}\quad LPI_{past}=\frac{10\times 1 + 5 \times 0.8}{10\times 1 + 5 \times 0.8} = \frac{14}{14}=1\] \[🧺_{past}|\text{🏷️}_{today}\quad LPI_{today}=\frac{10\times 1.2 + 5\times 1.1}{10\times 1 + 5 \times 0.8}=\frac{17.5}{14}=1.25\]
Remember that 1 corresponds to 100, and 1.25 corresponds to 125.
Kids’ Explanation - Prices
Paasche - Price:
Use always \(🧺_{today}\)! Paasche Price Index (PPI):
\[🧺_{today}|\text{🏷️}_{past}\quad PPI_{past}=\frac{8\times 1 + 6 \times 0.8}{8\times 1 + 6 \times 0.8} = \frac{12.8}{12.8}=1\] \[🧺_{today}|\text{🏷️}_{today}\quad PPI_{today}=\frac{8\times 1.2 + 6\times 1.1}{8\times 1 + 6 \times 0.8}=\frac{16.2}{12.8}=1.27\]
Remember that 1 corresponds to 100, and 1.27 corresponds to 127.
In general - Laspeyres
For prices, how much the old basket would cost at current prices relative to the old basket and prices? \[LPI = \frac{\sum_{k=1}^m p_t^k q_0^k}{\sum_{k=1}^m p_0^k q_0^k}\]
For quantities, how much the current basket would cost at the old prices, relative to the old basket and prices? \[LQI = \frac{\sum_{k=1}^m p_0^k q_t^k}{\sum_{k=1}^m p_0^k q_0^k}\]
Note: denominator is the same.
In general - Paasche
For prices, how much more expensive is the current basket at current prices, relative to the original prices. \[PPI = \frac{\sum_{k=1}^m p_t^kq_t^k}{\sum_{k=1}^m p_0^kq_t^k}\]
For quantities, how much more the current basket costs, relative to what would cost the original basket at current prices. \[PQI = \frac{\sum_{k=1}^m p_t^kq_t^k}{\sum_{k=1}^m p_t^kq_0^k}\]
Note: numerator is the same.
Conclusions
In both index numbers, for prices, quantities are constant: All variation comes from prices.
In both index numbers, for quantities, prices are constant: All variation comes from quantities.
We tend to use much more often Laspeyres because quantities take much longer to be accurately collected than prices. Ergo, national statistics institutes usually use Laspeyres price index to measure the price evolution.
Do the two indices agree?
Two goods, both costing 1 at the base period, both bought 10 times. Now good 1 gets more expensive, and people react by buying less of it.
Left slider: how much more expensive good 1 becomes. Right slider: how strongly people substitute away from it.
Price of good 1: 2.00×
Substitution: 0.50
What the chart showed
With no substitution the two indices are identical: the basket never changed, so it does not matter whose basket you weight with.
As substitution grows, Laspeyres stays put (it always weights with the old basket, which still contains a lot of the good that got dearer) while Paasche falls (it weights with the new basket, which has already moved away from that good).
So Laspeyres tends to overstate the rise in the cost of living, and Paasche tends to understate it. The truth is somewhere in between, which is exactly the gap the next index tries to close.
Fisher Index
Some authors argue, however, that the true measure for cost of life is in between both metrics. This is captured by the Fisher Index:
\[FPI_{t|0}=\sqrt{LPI_{t|0}\times PPI_{t|0}}\]
\[FQI_{t|0}=\sqrt{LQI_{t|0}\times PQI_{t|0}}\]
Example: the data
A basket of 15 goods, observed in 2024, 2025 and 2026. Each cell is what the basket costs with the row year’s prices and the column year’s quantities, \(\sum_{k=1}^{15} p_{row}^k\, q_{col}^k\).
| \(q_{2024}\) | \(q_{2025}\) | \(q_{2026}\) | |
|---|---|---|---|
| \(p_{2024}\) | 8562 | 9210 | 9920 |
| \(p_{2025}\) | 9150 | 9584 | |
| \(p_{2026}\) | 9657 | 10013 |
\[LPI_{2025|2024}=100\times \frac{\color{#8B1538}{9150}}{\color{#8B1538}{8562}}\]
\[PPI_{2026|2024}=100\times\frac{\color{#0F766E}{10013}}{\color{#0F766E}{9920}}\]
Example: LPI, PPI and FPI
What are the three indices if we choose 2024 as the base year?
| \(t\) | 2024 | 2025 | 2026 |
|---|---|---|---|
| \(LPI_{t\vert 2024}\) | 100 | 106.9 | 112.8 |
| \(PPI_{t\vert 2024}\) | 100 | 104.1 | 100.9 |
| \(FPI_{t\vert 2024}\) | 100 | 105.5 | 106.7 |
\[FPI_{2025|2024}=\color{#B8860B}{\sqrt{\color{#8B1538}{106.9}\times \color{#0F766E}{104.1}}}\]
❓ Composite Index Numbers · Question 1
The Laspeyres price index weights prices using quantities from:
A. the current period
B. the base period
C. the average of both
D. neither, it uses no quantities
✅ B. Laspeyres fixes the base period basket, \(LPI_{t|0}=\frac{\sum_k p_t^k q_0^k}{\sum_k p_0^k q_0^k}\). Paasche is the one that uses current quantities.
❓ Composite Index Numbers · Question 2
The Fisher index is:
A. the arithmetic mean of Laspeyres and Paasche
B. the geometric mean of Laspeyres and Paasche
C. always between 100 and 200
D. always equal to Laspeyres
✅ B. \(FPI_{t|0}=\sqrt{LPI_{t|0}\times PPI_{t|0}}\).
✏️ Composite Index Numbers · Question 3
For a basket you computed \(LPI_{t|0}=110\) and \(PPI_{t|0}=104\).
Compute Fisher’s price index, and explain why it must always fall between the other two.
✅ Composite Index Numbers · Solution
\(FPI_{t|0}=\sqrt{110\times 104}=\sqrt{11440}\approx 106.96\)
Fisher is the geometric mean of the other two, and the geometric mean of two positive numbers always lies between them. So whenever \(LPI\neq PPI\), Fisher sits strictly in between, here between 104 and 110.
Inflation
Consumer Price Index
To measure inflation, we use a Laspeyres Price Index, usually known as CPI (Consumer Price Index).
To do that, the National Institute of Statistics (or the equivalent national statistical office) defines a basket, which contains a certain mix of goods and services, that is to be representative of the consumption behavior of the population.
In Portugal :portugal:, the index has base in 2016, and, as stated above, follows a LPI formulation with small technical adjustments. This index has a monthly frequency, i.e. it is computed every month.
Consumer Price Index
To create a price index, you need to follow at least 3 steps.
- Determine representative basket: Survey families :family_woman_woman_girl_boy: :family_man_woman_boy_boy: :family_man_man_girl_boy:
- Survey retail stores to find out sale prices. :convenience_store:
- Compute the index with the collected data. :chart_with_upwards_trend:
- Compute the cost of the :basket: with the prices at the base period.
- Compute the cost of the same :basket: with the prices of every successive period.
Application
A Labor union leader is complaining that the average salary in 2022 was 1500, and in 2025 it is 1650. This increase, they say, is not enough to maintain the workers’ purchasing power, because of inflation. The data for inflation is the following:
| Year | 2022 | 2023 | 2024 | 2025 |
|---|---|---|---|---|
| CPI | 113 | 119 | 122.2 | 124.8 |
- The year with the largest increase in prices was 2024?
- The average growth rate for the wages between 2022 and 2025 was _______?
- Is the union leader right?
Application
For the first question, we can compute the chain or link index:
|
Year |
2022 |
2023 |
2024 |
2025 |
|---|---|---|---|---|
|
CPI |
113 |
119 |
122.2 |
124.8 |
|
CPIlink |
— |
105.3 |
102.7 |
102.1 |
So the answer is: The year with the largest price increase was 2023.
Just in case there were doubts, an example: \[CPI_{link|2024}=\frac{122.2}{119}\times 100=102.7\]
Application
We know salary in 2022 was 1500, and in 2025 1650. If the growth rate was constant, say \(\delta_w\) we would have:
\[1650 = 1500(1+\delta_w)^{3}\]
Or conversely
\[\frac{1650}{1500}=(1+\delta_w)^{3}\]
or
\[\left(1.1\right)^{1/3}=1+\delta_w\]
\[1.0323-1=\delta_w \quad \Rightarrow \quad \delta_w=0.0323= 3.23\%\]
Summarizing
Remember this relationship we saw before? \[I^q_{t|b}=\frac{I^v_{t|b}}{I^p_{t|b}}\]
Real equals nominal over CPI.
This, dividing by CPI a nominal quantity, is known as to deflate by the CPI.
About the Test for Factor Reversal for Composite Index Numbers
Note that neither Laspeyres nor Paasche pass the test for factor reversal (see here), \[VI_{t|0}\neq LPI_{t|0}\times LQI_{t|0}\quad\quad VI_{t|0}\neq PPI_{t|0}\times PQI_{t|0}\]
But we can verify that: \[LPI_{t|0}\times PQI_{t|0}=IV_{t|0} \quad \quad PPI_{t|0}\times LQI_{t|0}=IV_{t|0}\]
About the Test for Factor Reversal for Composite Index Numbers
(… continuation)
Example (do the rest as homework): \[LPI_{t|0}\times PQI_{t|0}=\frac{\sum_{k=1}^m p_t^kq_0^k}{\sum_{k=1}^m p_0^kq_0^k}\frac{\sum_{k=1}^m p_t^kq_t^k}{\sum_{k=1}^m p_t^kq_0^k}=\frac{\sum_{k=1}^m p_t^kq_t^k}{\sum_{k=1}^m p_0^kq_0^k}\]
Note that this last item is exactly \(IV_{t|0}\).
You can verify, using the same procedure, that Fisher’s index does indeed pass the factor reversal test.
About the Factor Reversal Test for Composite Index Numbers
\[FPI_{t|0}\times FQI_{t|0}=\] \[\sqrt{LPI_{t|0}\times PPI_{t|0}}\times\sqrt{LQI_{t|0}\times PQI_{t|0}}\] \[\sqrt{LPI_{t|0}\times PPI_{t|0}\times LQI_{t|0}\times PQI_{t|0}}\] \[\sqrt{LPI_{t|0}\times PQI_{t|0}\times LQI_{t|0}\times PPI_{t|0}}\] \[\sqrt{IV_{t|0}\times IV_{t|0}}=\sqrt{IV_{t|0}^2}=IV_{t|0}\]
About the Time Reversal Test for Composite Index Numbers
Laspeyres and Paasche do not pass the time Reversal test, let’s see an example:
\[LPI_{t|0}\times LPI_{0|t} = \frac{\sum_{k=1}^m p_t^k q_0^k}{\sum_{k=1}^m p_0^k q_0^k}\frac{\sum_{k=1}^m p_0^k q_t^k}{\sum_{k=1}^m p_t^k q_t^k}\neq 1\]
You can do the same for Paasche, and for the quantities index numbers.
About the Circularity Test for Composite Index Numbers
None, not even Fisher’s Index, passes the circularity test.
Homework: Verify it.
Example
| Public Transportation | Year 1 | Year 2 | ||
|---|---|---|---|---|
| Trips | Price per trip | Trips | Price Index (Base: year 1) | |
| Bus | 1,763,521 | 1.15 | 1,875,345 | 108.70 |
| Metro | 3,148,350 | 1.2 | 3,396,138 | 112.50 |
Example
- According to LPI for year 2, prices for public transportation grew 11.17%.
- Real change, according to Paasche, for public transportation was 7.35%.
- The composite index for change in expenditures with public transportation in year 2 compared the expenditure in year 1 is what type of change? (real, nominal, quantity, prices?) How much was it?
Example
According to LPI for year 2, prices for public transportation grew 11.17%.
First, we need to find \(p_2\) for bus and metro.
\[108.7=\frac{p_2^{bus}}{1.15}\times 100\]
Or \(p_2^{bus}=1.25\)
\[112.5=\frac{p_2^{metro}}{1.2}\]
Or \(p_2^{metro}=1.35\)
Example
|
|
Year 1 |
Year 2 |
|---|---|---|
|
Public Transportation |
N Trips |
Ticket price |
|
Bus |
1,763,521 |
1.15 |
|
Metro |
3,148,350 |
1.2 |
Example
\[LPI_{2|1}=\frac{\sum_{k=1}^m p_2^k q_1^k}{\sum_{k=1}^m p_1^k q_1^k}\]
\[LPI_{2|1}=\frac{1,763,521\times 1.25 + 3,148,350 \times 1.35}{1,763,521\times 1.15 + 3,148,350 \times 1.2}\]
\[LPI_{2|1}=1.1117\]
The basket in year 2 is 11.17% more expensive than in year 1. \[LPI_{2|1}\times 100 - 100 = 11.17\%\]
Example
Real change, according to Paasche, for public transportation was 7.35%.
\[PQI_{2|1}=\frac{\sum_{k=1}^m p_2^k q_2^k}{\sum_{k=1}^m p_2^k q_1^k}\]
\[PQI_{2|1}=\frac{1,875,345\times 1.25 + 3,396,138\times 1.35}{1,763,521\times 1.25 + 3,148,350\times 1.35}\]
\[PQI_{2|1}=1.0735\]
Indeed, according to the Paasche index, in year 2 the amount of public transportation trips was 7.35% larger than in year 1. \[PQI_{2|1}\times 100 - 100 = 7.35\%\]
Example
The composite index for change in expenditures with public transportation in year 2 compared the expenditure in year 1 is what type of change? (real, nominal, quantity, prices?) How much was it?
The value index for public transportation is a nominal index (price times quantity).
Remember: \[VI_{t|0}=LPI_{t|0}\times PQI_{t|0}\]
Example
\[VI_{2|1}=LPI_{2|1}\times PQI_{2|1} = 1.1117 \times 1.0735 \approx 1.1934\]
The value of public transportation trips grew 19.34% from year 1 to year 2. This includes the effect of an increase in prices and quantities.
❓ Inflation · Question 1
The Consumer Price Index is constructed as a:
A. Paasche quantity index
B. simple index number
C. Fisher index
D. Laspeyres price index
✅ D. The basket is fixed at a base period, which is exactly a Laspeyres price index (with small technical adjustments).
❓ Inflation · Question 2
Your nominal wage rises 10% while the CPI rises 12%. Your purchasing power:
A. rose about 10%
B. did not change
C. rose 2%
D. fell about 1.8%
✅ D. Real change \(=\frac{1.10}{1.12}-1\approx -0.0179\), a fall of about 1.8%. Prices grew faster than the wage.
✏️ Inflation · Question 3
A worker earned 1500 three years ago and earns 1650 today. Over the same period the CPI went from 100 to 112.
Did the worker gain or lose purchasing power, and by how much?
✅ Inflation · Solution
Nominal wage growth: \(\frac{1650}{1500}-1=0.10\), that is 10%. Price growth: \(\frac{112}{100}-1=0.12\), that is 12%.
\[\delta^q=\frac{1+\delta^v}{1+\delta^p}-1=\frac{1.10}{1.12}-1\approx -0.0179\]
The worker lost about 1.8% of purchasing power: the wage grew, but prices grew faster. ✅