7 Index Numbers

Status: ported 2026-05-19. Reviewed by editor: pending.

Learning outcomes

By the end of this chapter the reader should be able to:

Distinguish absolute variation, rate of variation, and unit variation (growth) factor, and compute each for a time series.
Compute the geometric-mean average growth rate and explain why the arithmetic mean of rates is biased upward.
Construct an elementary (simple) price index and use the circular and inversion properties to re-anchor it.
Compute the Laspeyres, Paasche, and Fisher composite indices for a basket of goods, and interpret each as a weighted average of elementary indices.
Apply the value index decomposition $V = L_P \cdot P_Q / 100 = P_P \cdot L_Q / 100$ to attribute changes in total expenditure to price and quantity components.
Link two index series with different base years and deflate a nominal monetary series to express it in constant (base-year) euros.
Use sub-group weights to compute an aggregated CPI and the contribution of each sub-group to overall inflation.
Diagnose the substitution-bias critique of Laspeyres-type cost-of-living measures and articulate why descriptive index changes do not identify causal effects.

Motivating empirical question

Is the average salary in Spain rising faster than prices — that is, are workers actually better off, or does a bigger paycheck just buy the same basket?

The Spanish Salario Mínimo Interprofesional (SMI) rose from EUR 736 per month in 2018 to EUR 1,134 in 2024, an apparent nominal increase of 54%. Over the same period the Índice de Precios al Consumo (IPC, the Spanish CPI) climbed from a level of 95.1 to 113.2, meaning the typical household basket became roughly 19% more expensive. To compare these magnitudes — and to answer the broader question of how purchasing power evolves — we need a coherent way to summarise the change of a single variable (a price, a wage, an output level) and, more ambitiously, the change of a bundle of variables (a consumer basket, a country’s industrial output, a stock index). Index numbers are the descriptive statistic that makes this possible. The same machinery will reappear in Chapter 7, where index series become the raw material for trend, cycle, and seasonal decomposition.

7.1 6.1 Introduction: what is an index number?

Definition: index number

An index number is a statistical measure that expresses the relative change of a variable (or a group of variables) with respect to a reference period, called the base period. Index numbers are conventionally scaled so that the base period equals 100.

Index numbers turn up everywhere a comparison across time or across countries is needed:

The Consumer Price Index (CPI / IPC) tracks the cost of a representative household basket. The Spanish IPC, published monthly by the INE, is the most widely cited inflation indicator.
The GDP deflator converts nominal GDP to real GDP, isolating the volume of production from price movements.
Stock-market indices such as the IBEX-35 (Spain), DAX (Germany), or S&P 500 (USA) summarise the price evolution of a basket of listed firms.
The Human Development Index combines health, education, and income into a single composite score, allowing cross-country comparison.

Before aggregating many goods, however, we need clean language for the change of a single variable. That is the next section.

7.2 6.2 Rates of variation and growth factors

Let $Y_t$ denote the value of a variable in period $t$.

Definition: absolute variation, rate of variation, growth factor

The absolute variation between $t-1$ and $t$ is

\[ \Delta_{t/t-1} = Y_t - Y_{t-1}, \]

measured in the same units as $Y$. The rate of variation (relative change, growth rate) is

\[ T_{t/t-1} = \frac{Y_t - Y_{t-1}}{Y_{t-1}} = \frac{Y_t}{Y_{t-1}} - 1, \]

and the unit variation factor (growth factor) is

\[ F_{t/t-1} = \frac{Y_t}{Y_{t-1}} = 1 + T_{t/t-1}. \]

A factor of $1.05$ means a 5% increase; a factor of $0.97$ means a 3% decrease.

7.2.1 6.2.1 Compounding: factors multiply, rates do not add

If a quantity grows over several periods, the overall change is the product of the per-period factors, not the sum of the per-period rates.

Common mistake: averaging growth by adding rates

It is tempting to summarise a multi-period growth experience by adding the period-by-period rates. Don’t. Rates compound multiplicatively. If a series grows by 10% one year and falls by 10% the next, it does not return to its starting value: $1.10 \times 0.90 = 0.99$, a 1% net decline.

Formally:

\[ F_{T/0} = \frac{Y_T}{Y_0} = \prod_{t=1}^{T} F_{t/t-1}, \qquad 1 + T_{T/0} = \prod_{t=1}^{T} (1 + T_{t/t-1}). \]

7.2.2 6.2.2 The average rate of variation is a geometric mean

The average rate of variation $\bar{T}$ over $T$ periods is the constant rate that, applied each period, reproduces the observed overall change:

\[ (1 + \bar{T})^T = \frac{Y_T}{Y_0} \quad\Longrightarrow\quad \bar{T} = \left(\frac{Y_T}{Y_0}\right)^{1/T} - 1. \]

Equivalently, $1 + \bar{T}$ is the geometric mean of the $T$ unit variation factors.

Example: consumption expenditure, Dec 2015 – Jun 2016

A household’s monthly consumption (EUR) is 1,800, 1,850, 1,830, 1,870, 1,910, 1,940, 1,980 from December 2015 through June 2016 (so $Y_0 = 1{,}800$, $Y_6 = 1{,}980$, $T = 6$ months). The overall six-month growth factor is $1{,}980 / 1{,}800 = 1.10$, a 10% rise. The average monthly rate is $(1.10)^{1/6} - 1 = 1.601\%$. The arithmetic mean of the six monthly rates is 1.61% — close, but biased upward.

7.3 6.3 Elementary index numbers

We now formalise the tracking of a single variable.

Definition: elementary index number

The elementary (or simple) index number of $Y$ in period $t$ with base period $0$ is

\[ I_{t/0} = \frac{Y_t}{Y_0} \times 100. \]

It equals 100 in the base period and reports the level as a percentage of the base. An index of 125 means the variable is 25% higher than in the base period; an index of 90 means 10% lower. The rate of variation between any two periods can be recovered as

\[ T_{t/0} = \frac{I_{t/0} - 100}{100}. \]

7.3.1 6.3.1 Properties: identity, inversion, circular (chain)

Properties of elementary indices

Let $I_{t/0}$ denote the elementary index of $Y$ in period $t$ with base $0$. Then:

Identity: $I_{0/0} = 100$.
Inversion: $I_{0/t} = 10{,}000 / I_{t/0}$.
Circular (chain) property: for any intermediate period $t'$, \[ I_{t/0} = \frac{I_{t/t'} \cdot I_{t'/0}}{100}, \qquad I_{t/t'} = \frac{I_{t/0}}{I_{t'/0}} \times 100. \]

All three follow directly from the definition $I_{a/b} = (Y_a / Y_b) \times 100$. The circular property is the engine of base-changing and linking, which we will use heavily in Section 6.7.

Example: chaining two indices

Suppose a product’s price index reads $I_{2015/2010} = 120$ and $I_{2020/2015} = 115$. Then by the circular property \[ I_{2020/2010} = \frac{115 \cdot 120}{100} = 138. \] The price in 2020 was 38% higher than in 2010, and the implied inverse is $I_{2010/2020} = 10{,}000 / 138 = 72.46$.

7.4 6.4 Composite (synthetic) indices: Laspeyres, Paasche, Fisher

In practice we rarely care about a single price. The CPI aggregates hundreds of items; a stock index aggregates dozens of firms. We need composite (synthetic) indices.

The two natural design choices for an aggregator are whether to weight and, if so, which period’s weights to use. Unweighted aggregators (the Sauerbeck mean of elementary indices and the Bradstreet–Dutot ratio of arithmetic means) treat all goods equally, which is unrealistic — a household spends a much larger share on housing than on matches. The serious composite indices are the weighted ones.

7.4.1 6.4.1 Laspeyres price index

Definition: Laspeyres price index

Let $p_{i,t}$ and $q_{i,t}$ denote price and quantity of good $i$ in period $t$. The Laspeyres price index weights elementary price relatives by base-period quantities:

\[ L_P^{t/0} = \frac{\sum_{i=1}^{n} p_{i,t}\, q_{i,0}}{\sum_{i=1}^{n} p_{i,0}\, q_{i,0}} \times 100. \]

Equivalently, it is the weighted arithmetic mean of elementary price indices with base-period expenditure shares as weights:

\[ L_P^{t/0} = \sum_{i=1}^{n} w_{i,0}\, I_{i}^{t/0}, \qquad w_{i,0} = \frac{p_{i,0}\, q_{i,0}}{\sum_j p_{j,0}\, q_{j,0}}. \]

The Spanish IPC (and most national CPIs) is a Laspeyres-type index whose basket is refreshed every five years or so from household budget surveys.

7.4.2 6.4.2 Paasche and Fisher price indices

Definition: Paasche and Fisher price indices

The Paasche price index uses current-period quantities as weights:

\[ P_P^{t/0} = \frac{\sum_{i=1}^{n} p_{i,t}\, q_{i,t}}{\sum_{i=1}^{n} p_{i,0}\, q_{i,t}} \times 100. \]

The Fisher price index is the geometric mean of Laspeyres and Paasche:

\[ F_P^{t/0} = \sqrt{L_P^{t/0} \cdot P_P^{t/0}}. \]

7.4.3 6.4.3 Substitution bias: which one tells the truth?

When relative prices change, consumers substitute away from goods whose price has risen most. This has three consequences worth memorising.

Laspeyres keeps the base-period basket fixed, so it ignores substitution. It typically overstates the true cost-of-living change.
Paasche uses the current basket, which already embeds substitution. It typically understates the true cost-of-living change.
Fisher splits the difference. When prices and quantities co-move negatively (the normal case) we have $P_P \le F_P \le L_P$. Fisher is sometimes called the ideal index because it satisfies the time-reversal and factor-reversal tests that Laspeyres and Paasche fail.

Common mistake: reading the IPC as a cost-of-living change

A 5% rise in the Laspeyres-type IPC does not mean the true cost of maintaining the same standard of living rose by 5%. Because the basket is held fixed, substitution toward cheaper goods is unobserved and the welfare-equivalent change is typically less than 5%.

7.4.4 6.4.4 Full worked example: three-good basket

A simple economy has three goods in 2020 (base) and 2024:

Good	$p_0$	$q_0$	$p_t$	$q_t$
Bread (kg)	1.20	200	1.80	170
Milk (L)	0.90	300	1.10	310
Rice (kg)	1.50	100	2.10	80

The four expenditure sums are $\sum p_0 q_0 = 660$, $\sum p_t q_0 = 900$, $\sum p_0 q_t = 603$, $\sum p_t q_t = 815$, so

\[ L_P = \frac{900}{660} \times 100 = 136.36, \quad P_P = \frac{815}{603} \times 100 = 135.16, \quad F_P = \sqrt{136.36 \cdot 135.16} = 135.75. \]

As expected $P_P < F_P < L_P$, reflecting that consumers substituted away from bread and rice (whose prices rose 50% and 40%) toward the relatively cheaper milk.

7.5 6.5 Quantity indices and the value decomposition

Price and quantity play symmetric roles in index construction. Swap them and we obtain quantity indices.

Definition: Laspeyres, Paasche, Fisher quantity indices

\[ L_Q^{t/0} = \frac{\sum p_{i,0}\, q_{i,t}}{\sum p_{i,0}\, q_{i,0}} \times 100, \qquad P_Q^{t/0} = \frac{\sum p_{i,t}\, q_{i,t}}{\sum p_{i,t}\, q_{i,0}} \times 100, \qquad F_Q^{t/0} = \sqrt{L_Q \cdot P_Q}. \]

A Laspeyres quantity index of 112 means that, valued at base-period prices, the quantity bundle is 12% larger than in the base period.

The value index measures the change in total expenditure:

\[ V^{t/0} = \frac{\sum p_{i,t}\, q_{i,t}}{\sum p_{i,0}\, q_{i,0}} \times 100. \]

Because $\sum p_t q_t$ contains both price and quantity movements, the value index decomposes cleanly:

Theorem: value index decomposition

For any basket,

\[ V^{t/0} = \frac{L_P^{t/0} \cdot P_Q^{t/0}}{100} = \frac{P_P^{t/0} \cdot L_Q^{t/0}}{100}. \]

The two cross-pairings ($L_P$ with $P_Q$, or $P_P$ with $L_Q$) are exact algebraic identities; the all-Laspeyres or all-Paasche pairings hold only approximately. A useful approximate decomposition uses Fisher on both sides: $V \approx F_P \cdot F_Q / 100$.

Example: continuing the three-good basket

Continuing from §6.4.4, $V = 815/660 \times 100 = 123.48$ and $P_Q = 815/900 \times 100 = 90.56$. Check: $L_P \cdot P_Q / 100 = 1.3636 \cdot 0.9056 \cdot 100 = 123.50$ (the residual 0.02 is pure rounding). Prices rose by 36% while real quantities fell by 9%, giving a net 23% rise in total expenditure.

7.6 6.6 Aggregating the CPI: sub-group weights and contributions

The IPC is not computed directly over all consumer goods. The INE first builds twelve sub-group indices (food, housing, transport, …), each a Laspeyres-type composite, and then aggregates them with expenditure-share weights:

\[ \text{CPI}_{t/0} = \sum_{g=1}^{G} w_g \cdot I_{g,t/0}, \qquad \sum_g w_g = 1. \]

The contribution of sub-group $g$ to the overall CPI change is

\[ C_g = w_g \times (I_{g,t/0} - 100), \qquad \sum_g C_g = \text{CPI}_{t/0} - 100. \]

This identity is the standard tool for explaining headline inflation: a 7%-of-basket transport sub-index up by 35% contributes $0.07 \times 35 = 2.45$ percentage points, regardless of what is happening elsewhere.

Example: who is driving overall inflation?

In a five-group toy CPI with weights (food 0.25, housing 0.30, transport 0.15, clothing 0.10, other 0.20) and 2024 sub-indices (128, 118, 135, 105, 110), the overall index is $0.25 \cdot 128 + \dots + 0.20 \cdot 110 = 120.15$. The largest contribution is food (7.0 pp) — even though transport had the largest price increase (35%), its small weight means it contributes only 5.25 pp.

7.7 6.7 Linking and re-basing series

Statistical agencies periodically refresh the basket and reset the base. Spain’s IPC has used bases 1983, 1992, 2001, 2006, 2011, 2016, and 2021. To build a long series we must link segments.

Definition: linking (rebasing)

To convert an index from base $b_1$ to a new base $b_2$:

\[ I_{t/b_2} = \frac{I_{t/b_1}}{I_{b_2/b_1}} \times 100. \]

This is just the circular property of §6.3.1 applied to a base change.

Example: chaining a CPI across two bases

Spanish CPI base 1992 reports $I_{2001/1992} = 132.5$; base 2001 reports $I_{2005/2001} = 115.4$. The 2005 index on base 1992 is then

\[ I_{2005/1992} = \frac{115.4 \cdot 132.5}{100} = 152.91. \]

Prices in 2005 were 52.9% higher than in 1992.

The same machinery works in either direction. To bring an old-base series onto a new base, divide by the bridging value; to push a new-base series backward, multiply by it.

7.8 6.8 Deflation: from nominal to real

When comparing monetary values across time, we must strip out the effect of price changes.

Definition: deflation

To deflate a nominal value $Y_t^{\text{nom}}$ using price index $I_{t/0}$ is to compute the real (constant-price) value

\[ Y_t^{\text{real}} = \frac{Y_t^{\text{nom}}}{I_{t/0} / 100} = \frac{Y_t^{\text{nom}}}{I_{t/0}} \times 100. \]

The real value is expressed in base-year euros — euros with the purchasing power of the base year.

A closely related quantity is the required salary that would have preserved purchasing power. If a worker earned $S_0$ in the base period, the salary needed in period $t$ to buy the same basket is

\[ S_t^{\text{required}} = S_0 \cdot \frac{I_{t/0}}{100}. \]

If actual $S_t < S_t^{\text{required}}$, purchasing power has been lost.

Example: nominal vs real GDP

For a country with nominal GDP (EUR bn) of 500, 520, 480, 530, 575, 610 and a GDP deflator (base 2018 = 100) of 100.0, 102.5, 103.8, 108.0, 116.5, 121.0 over 2018–2023, real GDP in 2018 euros is 500.0, 507.3, 462.4, 490.7, 493.6, 504.1. Nominal output grew 22%; real output grew only 0.8%. The 2020 recession (visible as the drop to 462.4) is invisible if one looks only at nominal figures.

7.9 6.9 A causality caveat

Index numbers are descriptive summaries. Two indices that move together — say, the Spanish SMI and the IPC — do not prove that one caused the other. Confounders (energy shocks, post-COVID demand, ECB monetary policy) can move both series jointly. Causal claims require either an explicit experiment or a credible identifying restriction, neither of which is supplied by the index machinery developed in this chapter. We will return to this point repeatedly in Chapter 7 and in TC2 / Econometrics I.

7.10 6.10 R Lab — A mini-CPI for a Granada student basket

The lab builds a five-good student basket (coffee, bus pass, rent, textbooks, beer), computes Laspeyres / Paasche / Fisher price and quantity indices, deflates a monthly allowance, links two CPI segments, and finally compares the Spanish minimum wage (SMI) against the IPC.

7.10.1 6.10.1 Setup

Code

set.seed(2026)
# No external libraries needed — base R is sufficient here.

7.10.2 6.10.2 Elementary index for a single good

Code

years  <- 2019:2024
coffee <- c(1.10, 1.15, 1.20, 1.30, 1.40, 1.55)
names(coffee) <- years

I_coffee <- coffee / coffee["2019"] * 100
round(I_coffee, 2)

  2019   2020   2021   2022   2023   2024 
100.00 104.55 109.09 118.18 127.27 140.91

Coffee prices have risen about 41% between 2019 and 2024.

7.10.3 6.10.3 Rates of variation and the geometric-mean average

Code

rov <- diff(coffee) / coffee[-length(coffee)] * 100
names(rov) <- paste0(years[-1], "/", years[-length(years)])
round(rov, 2)

2020/2019 2021/2020 2022/2021 2023/2022 2024/2023 
     4.55      4.35      8.33      7.69     10.71

Code

# Geometric-mean average annual rate
n_periods   <- length(coffee) - 1
total_ratio <- coffee[length(coffee)] / coffee[1]
avg_rate    <- (total_ratio^(1 / n_periods) - 1) * 100
round(avg_rate, 2)

2024 
 7.1

The geometric-mean average rate is the correct summary; the arithmetic mean of the annual rates would over-state the true growth.

7.10.4 6.10.4 Student basket — Laspeyres, Paasche, Fisher (price)

Code

goods <- c("Coffee", "Bus pass", "Rent", "Textbooks", "Beer")
p0 <- c(1.15, 40, 350, 45, 1.80)    # prices in 2020
p1 <- c(1.55, 52, 420, 50, 2.30)    # prices in 2024
q0 <- c(300, 10, 12, 6, 200)        # quantities in 2020
q1 <- c(280, 10, 12, 5, 180)        # quantities in 2024

L_P <- sum(p1 * q0) / sum(p0 * q0) * 100
P_P <- sum(p1 * q1) / sum(p0 * q1) * 100
F_P <- sqrt(L_P * P_P)
round(c(Laspeyres = L_P, Paasche = P_P, Fisher = F_P), 2)

Laspeyres   Paasche    Fisher 
    121.7     121.7     121.7

Prices for the student basket rose roughly 19–20% between 2020 and 2024. Fisher lies between Laspeyres and Paasche, as the substitution-bias story predicts.

7.10.5 6.10.5 Quantity indices and the value decomposition

Code

L_Q <- sum(q1 * p0) / sum(q0 * p0) * 100
P_Q <- sum(q1 * p1) / sum(q0 * p1) * 100
F_Q <- sqrt(L_Q * P_Q)
round(c(L_Q = L_Q, P_Q = P_Q, F_Q = F_Q), 2)

  L_Q   P_Q   F_Q 
98.13 98.13 98.13

Code

# Two ways to compute the value index
V_direct <- sum(p1 * q1) / sum(p0 * q0) * 100
V_decomp <- L_P * P_Q / 100
round(c(direct = V_direct, decomposition = V_decomp), 2)

       direct decomposition 
       119.43        119.43

The decomposition $V = L_P \cdot P_Q / 100$ holds exactly (up to floating-point error).

7.10.6 6.10.6 Deflation: nominal vs real student allowance

Code

years_d   <- 2020:2024
nominal   <- c(600, 620, 640, 670, 700)
price_idx <- c(100, 105.2, 110.8, 116.0, 122.6)
real      <- nominal / price_idx * 100

data.frame(year = years_d, nominal, price_idx, real = round(real, 2))

  year nominal price_idx   real
1 2020     600     100.0 600.00
2 2021     620     105.2 589.35
3 2022     640     110.8 577.62
4 2023     670     116.0 577.59
5 2024     700     122.6 570.96

Code

plot(years_d, nominal, type = "o", pch = 16, col = "steelblue",
     ylim = c(540, 720), lwd = 2,
     xlab = "Year", ylab = "EUR / month",
     main = "Nominal vs real student allowance")
lines(years_d, real, type = "o", pch = 17, col = "tomato", lwd = 2)
legend("topleft", legend = c("Nominal", "Real (2020 EUR)"),
       col = c("steelblue", "tomato"), pch = c(16, 17),
       lwd = 2, bty = "n")

Nominal vs real student allowance (real expressed in 2020 EUR).

The nominal allowance climbs from 600 to 700 EUR per month, but the real allowance (in 2020 EUR) falls to about 571 — purchasing power has declined because prices grew faster than the allowance.

7.10.7 6.10.7 Linking two CPI segments

Code

# Old series, base 2015 = 100, 2015–2019
old_base <- c(100, 101.5, 103.2, 105.0, 107.1)
names(old_base) <- 2015:2019

# New series, base 2019 = 100, 2019–2024
new_base <- c(100, 102.3, 105.1, 109.4, 112.6, 116.0)
names(new_base) <- 2019:2024

# Link the old series to the new base (2019 = 100)
link_factor   <- 100 / old_base["2019"]
old_linked    <- old_base * as.numeric(link_factor)

# Combine, dropping the duplicate 2019 from the old segment
linked_series <- c(old_linked[1:4], new_base)
round(linked_series, 2)

  2015   2016   2017   2018   2019   2020   2021   2022   2023   2024 
 93.37  94.77  96.36  98.04 100.00 102.30 105.10 109.40 112.60 116.00

The two segments now sit on a common base (2019 = 100) and can be analysed as one continuous series.

7.10.8 6.10.8 Real-data example: Spanish minimum wage vs IPC

Code

year_rd <- 2018:2024
smi     <- c(735.9, 900.0, 950.0, 965.0, 1000.0, 1080.0, 1134.0)
cpi_rd  <- c(95.1, 95.8, 95.5, 98.0, 106.1, 109.9, 113.2)

smi_idx  <- smi    / smi[1]    * 100   # SMI rebased to 2018 = 100
cpi_idx  <- cpi_rd / cpi_rd[1] * 100   # IPC rebased to 2018 = 100
real_smi <- smi_idx / cpi_idx * 100

round(data.frame(year = year_rd, smi_idx, cpi_idx, real_smi), 2)

  year smi_idx cpi_idx real_smi
1 2018  100.00  100.00   100.00
2 2019  122.30  100.74   121.41
3 2020  129.09  100.42   128.55
4 2021  131.13  103.05   127.25
5 2022  135.89  111.57   121.80
6 2023  146.76  115.56   127.00
7 2024  154.10  119.03   129.46

Code

plot(year_rd, smi_idx, type = "o", pch = 16, col = "steelblue",
     ylim = c(95, 160), lwd = 2,
     xlab = "Year", ylab = "Index (2018 = 100)",
     main = "SMI vs IPC, base 2018 = 100")
lines(year_rd, cpi_idx,  type = "o", pch = 17, col = "tomato", lwd = 2)
lines(year_rd, real_smi, type = "o", pch = 18, col = "darkgreen", lwd = 2)
legend("topleft",
       legend = c("Nominal SMI", "IPC", "Real SMI"),
       col = c("steelblue", "tomato", "darkgreen"),
       pch = c(16, 17, 18), lwd = 2, bty = "n")

Spanish minimum wage (SMI) vs IPC, both rebased to 2018 = 100, and the implied real SMI.

The real SMI rose substantially over 2018–2024 — nominal SMI growth outpaced inflation. Two warnings, both echoing §6.9: (i) the SMI is an instrument set by policy, not an outcome that should be deflated like a market wage; (ii) the co-movement of SMI and IPC does not identify a causal effect of SMI on inflation. Both series respond to broader macroeconomic forces.

Self-check

Q1. Reading an elementary index

An index for the price of coffee in 2024 (base 2019 = 100) equals 141. Which statement is correct?

A. Coffee was 141% more expensive in 2024 than in 2019.
B. Coffee in 2024 cost 41% more than in 2019.
C. Coffee in 2024 cost 1.41 EUR if it cost 1 EUR in 2019.
D. Coffee in 2024 cost 41 EUR more than in 2019.

Answer: B. The index 141 means the 2024 price is 141% of the 2019 price — equivalently, 41% higher. Option A confuses “of” with “more than”; D confuses index points with EUR. Option C is true arithmetically (the price of a 1 EUR coffee is now 1.41 EUR) but the percentage statement is the conventional interpretation.

Q2. Geometric vs arithmetic average growth

If a series goes from 100 (year 0) to 150 (year 5), the average annual rate of variation is:

A. $(150 - 100)/5 = 10\%$ per year (arithmetic mean of the increase).
B. $(1.50)^{1/5} - 1 \approx 8.45\%$ per year (geometric mean of factors).
C. $50 / 5 = 10\%$ per year, since the index is multiplicative.
D. Cannot be computed without the intermediate values.

Answer: B. Growth rates compound, so the constant rate that produces the observed five-fold-period increase is the geometric mean of the factors, $1.5^{1/5} \approx 1.0845$. Options A and C treat compounding as additive and over-state the rate.

Q3. Which weights does Laspeyres use?

The Laspeyres price index weights elementary price relatives by:

A. Base-period quantities $q_0$.
B. Current-period quantities $q_1$.
C. The arithmetic mean of $q_0$ and $q_1$.
D. The geometric mean of $q_0$ and $q_1$.

Answer: A. Laspeyres is the cost of the base-period basket at current prices, divided by the cost of the same basket at base-period prices. Paasche uses the current basket; Fisher takes a geometric mean of the two values, not of the quantities themselves.

Q4. Substitution bias

When relative prices change, consumers substitute away from goods whose price has risen most. As a result:

A. Laspeyres tends to overstate the cost-of-living change; Paasche tends to understate it.
B. Laspeyres tends to understate; Paasche tends to overstate.
C. Both indices are unbiased, since they use observed transactions.
D. The bias depends only on the base year, not on substitution.

Answer: A. Laspeyres holds the old basket fixed, ignoring substitution toward cheaper goods, and so over-states the cost increase. Paasche uses the new basket, which already reflects substitution, and so under-states it. Fisher splits the difference.

Q5. Deflating a wage

A worker’s nominal wage rises 20% while the CPI rises 30% over the same period. The real wage has:

A. Risen by about 20%.
B. Fallen by roughly 7.7% (since $1.20 / 1.30 - 1 \approx -0.077$).
C. Stayed constant — the increases roughly cancel out.
D. Risen, because nominal income grew.

Answer: B. Real growth equals the ratio of nominal growth to price growth, not their difference. The wage buys $1.20 / 1.30 \approx 92.3\%$ of the basket it used to buy.

Q6. Real GDP

If nominal GDP grows 4% and the GDP deflator grows 5%, real GDP:

A. Grows about 4%.
B. Falls by roughly 1% (real output shrank).
C. Grows about 9%.
D. Stays constant.

Answer: B. Real GDP equals nominal GDP divided by the deflator (×100); $1.04 / 1.05 \approx 0.990$, a 1% contraction in real terms. C confuses real growth with the sum of the two changes.

Q7. Linking two CPI segments

An old CPI (base 2010 = 100) reads 107.1 in 2019. A new CPI (base 2019 = 100) reads 116.0 in 2024. To link both into one continuous series with 2019 = 100:

A. Multiply the old series by $100/107.1$ and leave the new series unchanged.
B. Multiply the new series by $107.1 / 100$.
C. Take the arithmetic mean of the two series in 2019.
D. Subtract 7.1 from every old value.

Answer: A. Dividing the old series by 107.1 (and multiplying by 100) re-anchors it so that the bridging year 2019 reads 100 — exactly the value the new series already shows. Option B re-anchors the wrong segment.

Q8. Causality

Spain’s nominal Minimum Wage (SMI) and the IPC both rose between 2018 and 2024. A journalist writes: “Minimum-wage hikes caused inflation.” What is the most appropriate critique from this course?

A. The claim is correct because Laspeyres overstates inflation.
B. Both series trend upward, but co-movement of two indices over a few years cannot identify the causal effect of SMI on inflation. Confounders (energy shock, post-COVID demand, ECB policy) drive both series.
C. There is no problem — if both series went up, one must cause the other.
D. The claim is correct only if Fisher equals Paasche.

Answer: B. Index numbers are descriptive; co-movement is consistent with many causal stories, including third-factor confounding. Identifying the causal effect of a policy on inflation requires either an experiment or a credible identifying restriction — beyond the scope of TC1.

Exercises

7.10.9 Exercise 6.1 ★ — Quarterly revenue: absolute and relative rates

The quarterly revenue (thousand EUR) of a small business is Q1 = 120, Q2 = 138, Q3 = 150, Q4 = 162.

Compute the absolute variation between each pair of consecutive quarters.
Compute the rate of variation (growth rate) between each pair.
In which quarter was the growth rate highest?

Solution

$\Delta_{Q2} = 18$, $\Delta_{Q3} = 12$, $\Delta_{Q4} = 12$ (thousand EUR).
$T_{Q2} = 18/120 = 15.00\%$, $T_{Q3} = 12/138 = 8.70\%$, $T_{Q4} = 12/150 = 8.00\%$.
Q2 had the largest growth rate (15%), even though the absolute change in Q2 and Q3 was different by only 6 thousand EUR.

7.10.10 Exercise 6.2 ★ — Average growth from link factors

Annual production (tonnes) is 800, 856, 924, 980 over 2020–2023.

Compute the link factors $f_t = Y_t / Y_{t-1}$.
Compute the average annual growth rate as the geometric mean of the link factors.
Verify with $\bar{T} = (Y_T / Y_0)^{1/T} - 1$.

Solution

$f_{2021} = 856/800 = 1.0700$, $f_{2022} = 924/856 \approx 1.0794$, $f_{2023} = 980/924 \approx 1.0606$.
Product $\approx 1.2250$; cube root $\approx 1.0701$, so $\bar{T} \approx 7.01\%$.
$(980/800)^{1/3} - 1 = 1.2250^{1/3} - 1 \approx 7.01\%$. Confirmed.

7.10.11 Exercise 6.3 ★ — Elementary index and the circular property

The price (EUR/kg) of a commodity is 4.00 (2021), 4.60 (2022), 5.29 (2023).

Compute $I_{2022/2021}$ and $I_{2023/2021}$.
Compute $I_{2023/2022}$.
Verify the circular property $I_{2023/2021} = I_{2023/2022} \cdot I_{2022/2021} / 100$.

Solution

$I_{2022/2021} = (4.60/4.00) \cdot 100 = 115.00$; $I_{2023/2021} = (5.29/4.00) \cdot 100 = 132.25$.
$I_{2023/2022} = (5.29/4.60) \cdot 100 = 115.00$.
$115 \cdot 115 / 100 = 132.25$. Confirmed.

7.10.12 Exercise 6.4 ★★ — Laspeyres, Paasche, Fisher with four goods

A family consumes apples, chicken, olive oil, and pasta. In 2019: prices (EUR) 2.00, 5.50, 4.00, 1.20 and quantities 50, 30, 20, 40. In 2024: prices 2.80, 7.20, 8.50, 1.50 and quantities 40, 28, 12, 55.

Compute $L_P^{2024/2019}$, $P_P^{2024/2019}$, $F_P^{2024/2019}$.
Compute the value index $V^{2024/2019}$.
Verify the decomposition $V = L_P \cdot P_Q / 100$.
Identify the substitution effect from the data.

7.10.13 Exercise 6.5 ★★ — Linking Spanish CPI across a base change

Spain changed CPI base from 2016 to 2021. Available data: base 2016 reads 103.2 (2018), 104.0 (2019), 103.5 (2020), 106.8 (2021); base 2021 reads 100.0 (2021), 108.4 (2022), 112.1 (2023).

Express all values on base 2016 = 100.
Express all values on base 2021 = 100.
Compute cumulative inflation from 2018 to 2023.

7.10.14 Exercise 6.6 ★★ — Deflating a nominal wage series

Annual nominal salary (EUR) is 24,000, 24,600, 25,400, 26,800, 28,000 over 2019–2023. CPI (base 2019 = 100) is 100.0, 100.8, 103.5, 112.0, 116.2.

Compute the real salary (in 2019 EUR) for each year.
Compute cumulative nominal growth 2019 → 2023.
Compute cumulative real growth 2019 → 2023.
Comment on the worker’s purchasing-power trajectory.

7.10.15 Exercise 6.7 ★★ — Required salary and purchasing power

A Granada professor earned EUR 36,000 in 2019. Spanish CPI (base 2016 = 100) is 104.5 (2019) and 121.3 (2024).

Compute the salary in 2024 that would preserve 2019 purchasing power.
If the actual 2024 salary is EUR 39,500, has purchasing power risen or fallen?
Compute the percentage change in purchasing power.

7.10.16 Exercise 6.8 ★★★ — CPI aggregation and inflation drivers

A four-sub-group CPI has weights and indices in 2022 (= 100 across the board), 2023, and 2024:

Sub-group	Weight	CPI 2023	CPI 2024
Food	0.30	110.5	114.2
Housing	0.35	104.0	108.6
Transport	0.20	115.8	118.3
Other	0.15	102.5	105.0

Compute the overall CPI for 2023 and 2024.
Compute year-on-year inflation rates for 2023 and 2024.
Compute each sub-group’s contribution to 2023 inflation and identify the principal driver.
Has inflation accelerated or decelerated from 2023 to 2024? Comment in light of the substitution-bias warning in §6.4.3.

--- title: "Index Numbers" --- > *Status: ported 2026-05-19. Reviewed by editor: pending.* ## Learning outcomes {.unnumbered} By the end of this chapter the reader should be able to: - Distinguish absolute variation, rate of variation, and unit variation (growth) factor, and compute each for a time series. - Compute the **geometric-mean** average growth rate and explain why the arithmetic mean of rates is biased upward. - Construct an **elementary (simple) price index** and use the circular and inversion properties to re-anchor it. - Compute the **Laspeyres**, **Paasche**, and **Fisher** composite indices for a basket of goods, and interpret each as a weighted average of elementary indices. - Apply the **value index decomposition** $V = L_P \cdot P_Q / 100 = P_P \cdot L_Q / 100$ to attribute changes in total expenditure to price and quantity components. - **Link** two index series with different base years and **deflate** a nominal monetary series to express it in constant (base-year) euros. - Use sub-group weights to compute an aggregated CPI and the contribution of each sub-group to overall inflation. - Diagnose the **substitution-bias critique** of Laspeyres-type cost-of-living measures and articulate why descriptive index changes do not identify causal effects. ## Motivating empirical question {.unnumbered} > *Is the average salary in Spain rising faster than prices — that is, are workers actually better off, or does a bigger paycheck just buy the same basket?* The Spanish *Salario Mínimo Interprofesional* (SMI) rose from EUR 736 per month in 2018 to EUR 1,134 in 2024, an apparent nominal increase of 54%. Over the same period the *Índice de Precios al Consumo* (IPC, the Spanish CPI) climbed from a level of 95.1 to 113.2, meaning the typical household basket became roughly 19% more expensive. To compare these magnitudes — and to answer the broader question of how purchasing power evolves — we need a coherent way to summarise the change of a single variable (a price, a wage, an output level) and, more ambitiously, the change of a *bundle* of variables (a consumer basket, a country's industrial output, a stock index). Index numbers are the descriptive statistic that makes this possible. The same machinery will reappear in [Chapter 7](07-time-series.qmd), where index series become the raw material for trend, cycle, and seasonal decomposition. ## 6.1 Introduction: what is an index number? ::: {.callout-note} ## Definition: index number An **index number** is a statistical measure that expresses the relative change of a variable (or a group of variables) with respect to a **reference period**, called the **base period**. Index numbers are conventionally scaled so that the base period equals 100. ::: Index numbers turn up everywhere a comparison across time or across countries is needed: - The **Consumer Price Index (CPI / IPC)** tracks the cost of a representative household basket. The Spanish IPC, published monthly by the INE, is the most widely cited inflation indicator. - The **GDP deflator** converts nominal GDP to real GDP, isolating the volume of production from price movements. - **Stock-market indices** such as the IBEX-35 (Spain), DAX (Germany), or S&P 500 (USA) summarise the price evolution of a basket of listed firms. - The **Human Development Index** combines health, education, and income into a single composite score, allowing cross-country comparison. Before aggregating many goods, however, we need clean language for the change of a *single* variable. That is the next section. ## 6.2 Rates of variation and growth factors Let $Y_t$ denote the value of a variable in period $t$. ::: {.callout-note} ## Definition: absolute variation, rate of variation, growth factor ::: The **absolute variation** between $t-1$ and $t$ is $$ \Delta_{t/t-1} = Y_t - Y_{t-1}, $$ measured in the same units as $Y$. The **rate of variation** (relative change, growth rate) is $$ T_{t/t-1} = \frac{Y_t - Y_{t-1}}{Y_{t-1}} = \frac{Y_t}{Y_{t-1}} - 1, $$ and the **unit variation factor** (growth factor) is $$ F_{t/t-1} = \frac{Y_t}{Y_{t-1}} = 1 + T_{t/t-1}. $$ A factor of $1.05$ means a 5% increase; a factor of $0.97$ means a 3% decrease. ### 6.2.1 Compounding: factors multiply, rates do not add If a quantity grows over several periods, the overall change is the *product* of the per-period factors, not the sum of the per-period rates. ::: {.callout-warning} ## Common mistake: averaging growth by adding rates It is tempting to summarise a multi-period growth experience by *adding* the period-by-period rates. Don't. Rates compound multiplicatively. If a series grows by 10% one year and falls by 10% the next, it does *not* return to its starting value: $1.10 \times 0.90 = 0.99$, a 1% net decline. ::: Formally: $$ F_{T/0} = \frac{Y_T}{Y_0} = \prod_{t=1}^{T} F_{t/t-1}, \qquad 1 + T_{T/0} = \prod_{t=1}^{T} (1 + T_{t/t-1}). $$ ### 6.2.2 The average rate of variation is a geometric mean The **average rate of variation** $\bar{T}$ over $T$ periods is the constant rate that, applied each period, reproduces the observed overall change: $$ (1 + \bar{T})^T = \frac{Y_T}{Y_0} \quad\Longrightarrow\quad \bar{T} = \left(\frac{Y_T}{Y_0}\right)^{1/T} - 1. $$ Equivalently, $1 + \bar{T}$ is the **geometric mean** of the $T$ unit variation factors. ::: {.callout-note} ## Example: consumption expenditure, Dec 2015 – Jun 2016 A household's monthly consumption (EUR) is 1,800, 1,850, 1,830, 1,870, 1,910, 1,940, 1,980 from December 2015 through June 2016 (so $Y_0 = 1{,}800$, $Y_6 = 1{,}980$, $T = 6$ months). The overall six-month growth factor is $1{,}980 / 1{,}800 = 1.10$, a 10% rise. The average monthly rate is $(1.10)^{1/6} - 1 = 1.601\%$. The arithmetic mean of the six monthly rates is 1.61% — close, but biased upward. ::: ## 6.3 Elementary index numbers We now formalise the tracking of a single variable. ::: {.callout-note} ## Definition: elementary index number ::: The **elementary** (or **simple**) **index number** of $Y$ in period $t$ with base period $0$ is $$ I_{t/0} = \frac{Y_t}{Y_0} \times 100. $$ It equals 100 in the base period and reports the level as a percentage of the base. An index of 125 means the variable is 25% higher than in the base period; an index of 90 means 10% lower. The rate of variation between any two periods can be recovered as $$ T_{t/0} = \frac{I_{t/0} - 100}{100}. $$ ### 6.3.1 Properties: identity, inversion, circular (chain) {#sec-index-properties} ::: {.callout-note} ## Properties of elementary indices Let $I_{t/0}$ denote the elementary index of $Y$ in period $t$ with base $0$. Then: 1. **Identity**: $I_{0/0} = 100$. 2. **Inversion**: $I_{0/t} = 10{,}000 / I_{t/0}$. 3. **Circular (chain) property**: for any intermediate period $t'$, $$ I_{t/0} = \frac{I_{t/t'} \cdot I_{t'/0}}{100}, \qquad I_{t/t'} = \frac{I_{t/0}}{I_{t'/0}} \times 100. $$ ::: All three follow directly from the definition $I_{a/b} = (Y_a / Y_b) \times 100$. The circular property is the engine of base-changing and linking, which we will use heavily in Section 6.7. ::: {.callout-note} ## Example: chaining two indices Suppose a product's price index reads $I_{2015/2010} = 120$ and $I_{2020/2015} = 115$. Then by the circular property $$ I_{2020/2010} = \frac{115 \cdot 120}{100} = 138. $$ The price in 2020 was 38% higher than in 2010, and the implied inverse is $I_{2010/2020} = 10{,}000 / 138 = 72.46$. ::: ## 6.4 Composite (synthetic) indices: Laspeyres, Paasche, Fisher In practice we rarely care about a single price. The CPI aggregates hundreds of items; a stock index aggregates dozens of firms. We need **composite (synthetic) indices**. The two natural design choices for an aggregator are *whether to weight* and, if so, *which period's weights to use*. Unweighted aggregators (the Sauerbeck mean of elementary indices and the Bradstreet–Dutot ratio of arithmetic means) treat all goods equally, which is unrealistic — a household spends a much larger share on housing than on matches. The serious composite indices are the weighted ones. ### 6.4.1 Laspeyres price index ::: {.callout-note} ## Definition: Laspeyres price index ::: Let $p_{i,t}$ and $q_{i,t}$ denote price and quantity of good $i$ in period $t$. The **Laspeyres price index** weights elementary price relatives by **base-period quantities**: $$ L_P^{t/0} = \frac{\sum_{i=1}^{n} p_{i,t}\, q_{i,0}}{\sum_{i=1}^{n} p_{i,0}\, q_{i,0}} \times 100. $$ Equivalently, it is the weighted arithmetic mean of elementary price indices with **base-period expenditure shares** as weights: $$ L_P^{t/0} = \sum_{i=1}^{n} w_{i,0}\, I_{i}^{t/0}, \qquad w_{i,0} = \frac{p_{i,0}\, q_{i,0}}{\sum_j p_{j,0}\, q_{j,0}}. $$ The Spanish IPC (and most national CPIs) is a Laspeyres-type index whose basket is refreshed every five years or so from household budget surveys. ### 6.4.2 Paasche and Fisher price indices ::: {.callout-note} ## Definition: Paasche and Fisher price indices ::: The **Paasche price index** uses **current-period quantities** as weights: $$ P_P^{t/0} = \frac{\sum_{i=1}^{n} p_{i,t}\, q_{i,t}}{\sum_{i=1}^{n} p_{i,0}\, q_{i,t}} \times 100. $$ The **Fisher price index** is the geometric mean of Laspeyres and Paasche: $$ F_P^{t/0} = \sqrt{L_P^{t/0} \cdot P_P^{t/0}}. $$ ### 6.4.3 Substitution bias: which one tells the truth? When relative prices change, consumers substitute *away* from goods whose price has risen most. This has three consequences worth memorising. - **Laspeyres** keeps the base-period basket fixed, so it ignores substitution. It typically **overstates** the true cost-of-living change. - **Paasche** uses the current basket, which already embeds substitution. It typically **understates** the true cost-of-living change. - **Fisher** splits the difference. When prices and quantities co-move negatively (the normal case) we have $P_P \le F_P \le L_P$. Fisher is sometimes called the *ideal* index because it satisfies the time-reversal and factor-reversal tests that Laspeyres and Paasche fail. ::: {.callout-warning} ## Common mistake: reading the IPC as a cost-of-living change A 5% rise in the Laspeyres-type IPC does **not** mean the true cost of maintaining the same standard of living rose by 5%. Because the basket is held fixed, substitution toward cheaper goods is unobserved and the welfare-equivalent change is typically *less* than 5%. ::: ### 6.4.4 Full worked example: three-good basket A simple economy has three goods in 2020 (base) and 2024: | Good | $p_0$ | $q_0$ | $p_t$ | $q_t$ | |-------------|------:|------:|------:|------:| | Bread (kg) | 1.20 | 200 | 1.80 | 170 | | Milk (L) | 0.90 | 300 | 1.10 | 310 | | Rice (kg) | 1.50 | 100 | 2.10 | 80 | The four expenditure sums are $\sum p_0 q_0 = 660$, $\sum p_t q_0 = 900$, $\sum p_0 q_t = 603$, $\sum p_t q_t = 815$, so $$ L_P = \frac{900}{660} \times 100 = 136.36, \quad P_P = \frac{815}{603} \times 100 = 135.16, \quad F_P = \sqrt{136.36 \cdot 135.16} = 135.75. $$ As expected $P_P < F_P < L_P$, reflecting that consumers substituted away from bread and rice (whose prices rose 50% and 40%) toward the relatively cheaper milk. ## 6.5 Quantity indices and the value decomposition Price and quantity play symmetric roles in index construction. Swap them and we obtain **quantity indices**. ::: {.callout-note} ## Definition: Laspeyres, Paasche, Fisher quantity indices ::: $$ L_Q^{t/0} = \frac{\sum p_{i,0}\, q_{i,t}}{\sum p_{i,0}\, q_{i,0}} \times 100, \qquad P_Q^{t/0} = \frac{\sum p_{i,t}\, q_{i,t}}{\sum p_{i,t}\, q_{i,0}} \times 100, \qquad F_Q^{t/0} = \sqrt{L_Q \cdot P_Q}. $$ A Laspeyres quantity index of 112 means that, *valued at base-period prices*, the quantity bundle is 12% larger than in the base period. The **value index** measures the change in total expenditure: $$ V^{t/0} = \frac{\sum p_{i,t}\, q_{i,t}}{\sum p_{i,0}\, q_{i,0}} \times 100. $$ Because $\sum p_t q_t$ contains both price and quantity movements, the value index decomposes cleanly: ::: {.callout-note} ## Theorem: value index decomposition For any basket, $$ V^{t/0} = \frac{L_P^{t/0} \cdot P_Q^{t/0}}{100} = \frac{P_P^{t/0} \cdot L_Q^{t/0}}{100}. $$ ::: The two cross-pairings ($L_P$ with $P_Q$, or $P_P$ with $L_Q$) are exact algebraic identities; the all-Laspeyres or all-Paasche pairings hold only approximately. A useful approximate decomposition uses Fisher on both sides: $V \approx F_P \cdot F_Q / 100$. ::: {.callout-note} ## Example: continuing the three-good basket Continuing from §6.4.4, $V = 815/660 \times 100 = 123.48$ and $P_Q = 815/900 \times 100 = 90.56$. Check: $L_P \cdot P_Q / 100 = 1.3636 \cdot 0.9056 \cdot 100 = 123.50$ (the residual 0.02 is pure rounding). Prices rose by 36% while real quantities *fell* by 9%, giving a net 23% rise in total expenditure. ::: ## 6.6 Aggregating the CPI: sub-group weights and contributions {#sec-cpi-aggregation} The IPC is not computed directly over all consumer goods. The INE first builds twelve sub-group indices (food, housing, transport, …), each a Laspeyres-type composite, and then aggregates them with expenditure-share weights: $$ \text{CPI}_{t/0} = \sum_{g=1}^{G} w_g \cdot I_{g,t/0}, \qquad \sum_g w_g = 1. $$ The **contribution** of sub-group $g$ to the overall CPI change is $$ C_g = w_g \times (I_{g,t/0} - 100), \qquad \sum_g C_g = \text{CPI}_{t/0} - 100. $$ This identity is the standard tool for explaining headline inflation: a 7%-of-basket transport sub-index up by 35% contributes $0.07 \times 35 = 2.45$ percentage points, regardless of what is happening elsewhere. ::: {.callout-note} ## Example: who is driving overall inflation? In a five-group toy CPI with weights (food 0.25, housing 0.30, transport 0.15, clothing 0.10, other 0.20) and 2024 sub-indices (128, 118, 135, 105, 110), the overall index is $0.25 \cdot 128 + \dots + 0.20 \cdot 110 = 120.15$. The largest *contribution* is food (7.0 pp) — even though transport had the largest *price increase* (35%), its small weight means it contributes only 5.25 pp. ::: ## 6.7 Linking and re-basing series Statistical agencies periodically refresh the basket and reset the base. Spain's IPC has used bases 1983, 1992, 2001, 2006, 2011, 2016, and 2021. To build a long series we must **link** segments. ::: {.callout-note} ## Definition: linking (rebasing) ::: To convert an index from base $b_1$ to a new base $b_2$: $$ I_{t/b_2} = \frac{I_{t/b_1}}{I_{b_2/b_1}} \times 100. $$ This is just the circular property of §6.3.1 applied to a base change. ::: {.callout-note} ## Example: chaining a CPI across two bases Spanish CPI base 1992 reports $I_{2001/1992} = 132.5$; base 2001 reports $I_{2005/2001} = 115.4$. The 2005 index on base 1992 is then $$ I_{2005/1992} = \frac{115.4 \cdot 132.5}{100} = 152.91. $$ Prices in 2005 were 52.9% higher than in 1992. ::: The same machinery works in either direction. To bring an old-base series onto a new base, divide by the bridging value; to push a new-base series backward, multiply by it. ## 6.8 Deflation: from nominal to real When comparing monetary values across time, we must strip out the effect of price changes. ::: {.callout-note} ## Definition: deflation ::: To **deflate** a nominal value $Y_t^{\text{nom}}$ using price index $I_{t/0}$ is to compute the **real (constant-price) value** $$ Y_t^{\text{real}} = \frac{Y_t^{\text{nom}}}{I_{t/0} / 100} = \frac{Y_t^{\text{nom}}}{I_{t/0}} \times 100. $$ The real value is expressed *in base-year euros* — euros with the purchasing power of the base year. A closely related quantity is the **required salary** that would have preserved purchasing power. If a worker earned $S_0$ in the base period, the salary needed in period $t$ to buy the same basket is $$ S_t^{\text{required}} = S_0 \cdot \frac{I_{t/0}}{100}. $$ If actual $S_t < S_t^{\text{required}}$, purchasing power has been lost. ::: {.callout-note} ## Example: nominal vs real GDP For a country with nominal GDP (EUR bn) of 500, 520, 480, 530, 575, 610 and a GDP deflator (base 2018 = 100) of 100.0, 102.5, 103.8, 108.0, 116.5, 121.0 over 2018–2023, real GDP in 2018 euros is 500.0, 507.3, 462.4, 490.7, 493.6, 504.1. Nominal output grew 22%; *real* output grew only 0.8%. The 2020 recession (visible as the drop to 462.4) is invisible if one looks only at nominal figures. ::: ## 6.9 A causality caveat Index numbers are descriptive summaries. Two indices that move together — say, the Spanish SMI and the IPC — do **not** prove that one caused the other. Confounders (energy shocks, post-COVID demand, ECB monetary policy) can move both series jointly. Causal claims require either an explicit experiment or a credible identifying restriction, neither of which is supplied by the index machinery developed in this chapter. We will return to this point repeatedly in [Chapter 7](07-time-series.qmd) and in TC2 / Econometrics I. ## 6.10 R Lab — A mini-CPI for a Granada student basket The lab builds a five-good *student basket* (coffee, bus pass, rent, textbooks, beer), computes Laspeyres / Paasche / Fisher price *and* quantity indices, deflates a monthly allowance, links two CPI segments, and finally compares the Spanish minimum wage (SMI) against the IPC. ### 6.10.1 Setup ```{r ch06-setup} #| message: false #| warning: false set.seed(2026) # No external libraries needed — base R is sufficient here. ``` ### 6.10.2 Elementary index for a single good ```{r ch06-coffee} years <- 2019:2024 coffee <- c(1.10, 1.15, 1.20, 1.30, 1.40, 1.55) names(coffee) <- years I_coffee <- coffee / coffee["2019"] * 100 round(I_coffee, 2) ``` Coffee prices have risen about 41% between 2019 and 2024. ### 6.10.3 Rates of variation and the geometric-mean average ```{r ch06-rates} rov <- diff(coffee) / coffee[-length(coffee)] * 100 names(rov) <- paste0(years[-1], "/", years[-length(years)]) round(rov, 2) # Geometric-mean average annual rate n_periods <- length(coffee) - 1 total_ratio <- coffee[length(coffee)] / coffee[1] avg_rate <- (total_ratio^(1 / n_periods) - 1) * 100 round(avg_rate, 2) ``` The geometric-mean average rate is the correct summary; the arithmetic mean of the annual rates would over-state the true growth. ### 6.10.4 Student basket — Laspeyres, Paasche, Fisher (price) ```{r ch06-basket} goods <- c("Coffee", "Bus pass", "Rent", "Textbooks", "Beer") p0 <- c(1.15, 40, 350, 45, 1.80) # prices in 2020 p1 <- c(1.55, 52, 420, 50, 2.30) # prices in 2024 q0 <- c(300, 10, 12, 6, 200) # quantities in 2020 q1 <- c(280, 10, 12, 5, 180) # quantities in 2024 L_P <- sum(p1 * q0) / sum(p0 * q0) * 100 P_P <- sum(p1 * q1) / sum(p0 * q1) * 100 F_P <- sqrt(L_P * P_P) round(c(Laspeyres = L_P, Paasche = P_P, Fisher = F_P), 2) ``` Prices for the student basket rose roughly 19–20% between 2020 and 2024. Fisher lies between Laspeyres and Paasche, as the substitution-bias story predicts. ### 6.10.5 Quantity indices and the value decomposition ```{r ch06-quantity} L_Q <- sum(q1 * p0) / sum(q0 * p0) * 100 P_Q <- sum(q1 * p1) / sum(q0 * p1) * 100 F_Q <- sqrt(L_Q * P_Q) round(c(L_Q = L_Q, P_Q = P_Q, F_Q = F_Q), 2) # Two ways to compute the value index V_direct <- sum(p1 * q1) / sum(p0 * q0) * 100 V_decomp <- L_P * P_Q / 100 round(c(direct = V_direct, decomposition = V_decomp), 2) ``` The decomposition $V = L_P \cdot P_Q / 100$ holds exactly (up to floating-point error). ### 6.10.6 Deflation: nominal vs real student allowance ```{r ch06-deflation} years_d <- 2020:2024 nominal <- c(600, 620, 640, 670, 700) price_idx <- c(100, 105.2, 110.8, 116.0, 122.6) real <- nominal / price_idx * 100 data.frame(year = years_d, nominal, price_idx, real = round(real, 2)) ``` ```{r ch06-deflation-plot} #| fig-width: 7 #| fig-height: 4.5 #| fig-cap: "Nominal vs real student allowance (real expressed in 2020 EUR)." plot(years_d, nominal, type = "o", pch = 16, col = "steelblue", ylim = c(540, 720), lwd = 2, xlab = "Year", ylab = "EUR / month", main = "Nominal vs real student allowance") lines(years_d, real, type = "o", pch = 17, col = "tomato", lwd = 2) legend("topleft", legend = c("Nominal", "Real (2020 EUR)"), col = c("steelblue", "tomato"), pch = c(16, 17), lwd = 2, bty = "n") ``` The nominal allowance climbs from 600 to 700 EUR per month, but the real allowance (in 2020 EUR) falls to about 571 — purchasing power has *declined* because prices grew faster than the allowance. ### 6.10.7 Linking two CPI segments ```{r ch06-linking} # Old series, base 2015 = 100, 2015–2019 old_base <- c(100, 101.5, 103.2, 105.0, 107.1) names(old_base) <- 2015:2019 # New series, base 2019 = 100, 2019–2024 new_base <- c(100, 102.3, 105.1, 109.4, 112.6, 116.0) names(new_base) <- 2019:2024 # Link the old series to the new base (2019 = 100) link_factor <- 100 / old_base["2019"] old_linked <- old_base * as.numeric(link_factor) # Combine, dropping the duplicate 2019 from the old segment linked_series <- c(old_linked[1:4], new_base) round(linked_series, 2) ``` The two segments now sit on a common base (2019 = 100) and can be analysed as one continuous series. ### 6.10.8 Real-data example: Spanish minimum wage vs IPC ```{r ch06-smi} year_rd <- 2018:2024 smi <- c(735.9, 900.0, 950.0, 965.0, 1000.0, 1080.0, 1134.0) cpi_rd <- c(95.1, 95.8, 95.5, 98.0, 106.1, 109.9, 113.2) smi_idx <- smi / smi[1] * 100 # SMI rebased to 2018 = 100 cpi_idx <- cpi_rd / cpi_rd[1] * 100 # IPC rebased to 2018 = 100 real_smi <- smi_idx / cpi_idx * 100 round(data.frame(year = year_rd, smi_idx, cpi_idx, real_smi), 2) ``` ```{r ch06-smi-plot} #| fig-width: 7 #| fig-height: 4.5 #| fig-cap: "Spanish minimum wage (SMI) vs IPC, both rebased to 2018 = 100, and the implied real SMI." plot(year_rd, smi_idx, type = "o", pch = 16, col = "steelblue", ylim = c(95, 160), lwd = 2, xlab = "Year", ylab = "Index (2018 = 100)", main = "SMI vs IPC, base 2018 = 100") lines(year_rd, cpi_idx, type = "o", pch = 17, col = "tomato", lwd = 2) lines(year_rd, real_smi, type = "o", pch = 18, col = "darkgreen", lwd = 2) legend("topleft", legend = c("Nominal SMI", "IPC", "Real SMI"), col = c("steelblue", "tomato", "darkgreen"), pch = c(16, 17, 18), lwd = 2, bty = "n") ``` The real SMI rose substantially over 2018–2024 — nominal SMI growth outpaced inflation. Two warnings, both echoing §6.9: (i) the SMI is an *instrument* set by policy, not an outcome that should be deflated like a market wage; (ii) the co-movement of SMI and IPC does not identify a causal effect of SMI on inflation. Both series respond to broader macroeconomic forces. ## Self-check {.unnumbered} ::: {.callout-tip collapse="true"} ## Q1. Reading an elementary index An index for the price of coffee in 2024 (base 2019 = 100) equals 141. Which statement is correct? - A. Coffee was 141% more expensive in 2024 than in 2019. - B. Coffee in 2024 cost 41% more than in 2019. - C. Coffee in 2024 cost 1.41 EUR if it cost 1 EUR in 2019. - D. Coffee in 2024 cost 41 EUR more than in 2019. **Answer: B.** The index 141 means the 2024 price is 141% *of* the 2019 price — equivalently, 41% higher. Option A confuses "of" with "more than"; D confuses index points with EUR. Option C is true arithmetically (the price of a 1 EUR coffee is now 1.41 EUR) but the percentage statement is the conventional interpretation. ::: ::: {.callout-tip collapse="true"} ## Q2. Geometric vs arithmetic average growth If a series goes from 100 (year 0) to 150 (year 5), the average annual rate of variation is: - A. $(150 - 100)/5 = 10\%$ per year (arithmetic mean of the increase). - B. $(1.50)^{1/5} - 1 \approx 8.45\%$ per year (geometric mean of factors). - C. $50 / 5 = 10\%$ per year, since the index is multiplicative. - D. Cannot be computed without the intermediate values. **Answer: B.** Growth rates compound, so the constant rate that produces the observed five-fold-period increase is the geometric mean of the factors, $1.5^{1/5} \approx 1.0845$. Options A and C treat compounding as additive and over-state the rate. ::: ::: {.callout-tip collapse="true"} ## Q3. Which weights does Laspeyres use? The Laspeyres price index weights elementary price relatives by: - A. Base-period quantities $q_0$. - B. Current-period quantities $q_1$. - C. The arithmetic mean of $q_0$ and $q_1$. - D. The geometric mean of $q_0$ and $q_1$. **Answer: A.** Laspeyres is the cost of the *base-period basket* at current prices, divided by the cost of the same basket at base-period prices. Paasche uses the current basket; Fisher takes a geometric mean of the two values, not of the quantities themselves. ::: ::: {.callout-tip collapse="true"} ## Q4. Substitution bias When relative prices change, consumers substitute away from goods whose price has risen most. As a result: - A. Laspeyres tends to overstate the cost-of-living change; Paasche tends to understate it. - B. Laspeyres tends to understate; Paasche tends to overstate. - C. Both indices are unbiased, since they use observed transactions. - D. The bias depends only on the base year, not on substitution. **Answer: A.** Laspeyres holds the old basket fixed, ignoring substitution toward cheaper goods, and so over-states the cost increase. Paasche uses the new basket, which already reflects substitution, and so under-states it. Fisher splits the difference. ::: ::: {.callout-tip collapse="true"} ## Q5. Deflating a wage A worker's nominal wage rises 20% while the CPI rises 30% over the same period. The real wage has: - A. Risen by about 20%. - B. Fallen by roughly 7.7% (since $1.20 / 1.30 - 1 \approx -0.077$). - C. Stayed constant — the increases roughly cancel out. - D. Risen, because nominal income grew. **Answer: B.** Real growth equals the *ratio* of nominal growth to price growth, not their difference. The wage buys $1.20 / 1.30 \approx 92.3\%$ of the basket it used to buy. ::: ::: {.callout-tip collapse="true"} ## Q6. Real GDP If nominal GDP grows 4% and the GDP deflator grows 5%, real GDP: - A. Grows about 4%. - B. Falls by roughly 1% (real output shrank). - C. Grows about 9%. - D. Stays constant. **Answer: B.** Real GDP equals nominal GDP divided by the deflator (×100); $1.04 / 1.05 \approx 0.990$, a 1% contraction in real terms. C confuses real growth with the sum of the two changes. ::: ::: {.callout-tip collapse="true"} ## Q7. Linking two CPI segments An old CPI (base 2010 = 100) reads 107.1 in 2019. A new CPI (base 2019 = 100) reads 116.0 in 2024. To link both into one continuous series with 2019 = 100: - A. Multiply the old series by $100/107.1$ and leave the new series unchanged. - B. Multiply the new series by $107.1 / 100$. - C. Take the arithmetic mean of the two series in 2019. - D. Subtract 7.1 from every old value. **Answer: A.** Dividing the old series by 107.1 (and multiplying by 100) re-anchors it so that the bridging year 2019 reads 100 — exactly the value the new series already shows. Option B re-anchors the wrong segment. ::: ::: {.callout-tip collapse="true"} ## Q8. Causality Spain's nominal Minimum Wage (SMI) and the IPC both rose between 2018 and 2024. A journalist writes: *"Minimum-wage hikes caused inflation."* What is the most appropriate critique from this course? - A. The claim is correct because Laspeyres overstates inflation. - B. Both series trend upward, but co-movement of two indices over a few years cannot identify the causal effect of SMI on inflation. Confounders (energy shock, post-COVID demand, ECB policy) drive both series. - C. There is no problem — if both series went up, one must cause the other. - D. The claim is correct only if Fisher equals Paasche. **Answer: B.** Index numbers are descriptive; co-movement is consistent with many causal stories, including third-factor confounding. Identifying the causal effect of a policy on inflation requires either an experiment or a credible identifying restriction — beyond the scope of TC1. ::: ## Exercises {.unnumbered} ### Exercise 6.1 ★ — Quarterly revenue: absolute and relative rates The quarterly revenue (thousand EUR) of a small business is Q1 = 120, Q2 = 138, Q3 = 150, Q4 = 162. (a) Compute the absolute variation between each pair of consecutive quarters. (b) Compute the rate of variation (growth rate) between each pair. (c) In which quarter was the growth rate highest? ::: {.callout-tip collapse="true"} ## Solution (a) $\Delta_{Q2} = 18$, $\Delta_{Q3} = 12$, $\Delta_{Q4} = 12$ (thousand EUR). (b) $T_{Q2} = 18/120 = 15.00\%$, $T_{Q3} = 12/138 = 8.70\%$, $T_{Q4} = 12/150 = 8.00\%$. (c) Q2 had the largest growth rate (15%), even though the absolute change in Q2 and Q3 was different by only 6 thousand EUR. ::: ### Exercise 6.2 ★ — Average growth from link factors Annual production (tonnes) is 800, 856, 924, 980 over 2020–2023. (a) Compute the link factors $f_t = Y_t / Y_{t-1}$. (b) Compute the average annual growth rate as the geometric mean of the link factors. (c) Verify with $\bar{T} = (Y_T / Y_0)^{1/T} - 1$. ::: {.callout-tip collapse="true"} ## Solution (a) $f_{2021} = 856/800 = 1.0700$, $f_{2022} = 924/856 \approx 1.0794$, $f_{2023} = 980/924 \approx 1.0606$. (b) Product $\approx 1.2250$; cube root $\approx 1.0701$, so $\bar{T} \approx 7.01\%$. (c) $(980/800)^{1/3} - 1 = 1.2250^{1/3} - 1 \approx 7.01\%$. Confirmed. ::: ### Exercise 6.3 ★ — Elementary index and the circular property The price (EUR/kg) of a commodity is 4.00 (2021), 4.60 (2022), 5.29 (2023). (a) Compute $I_{2022/2021}$ and $I_{2023/2021}$. (b) Compute $I_{2023/2022}$. (c) Verify the circular property $I_{2023/2021} = I_{2023/2022} \cdot I_{2022/2021} / 100$. ::: {.callout-tip collapse="true"} ## Solution (a) $I_{2022/2021} = (4.60/4.00) \cdot 100 = 115.00$; $I_{2023/2021} = (5.29/4.00) \cdot 100 = 132.25$. (b) $I_{2023/2022} = (5.29/4.60) \cdot 100 = 115.00$. (c) $115 \cdot 115 / 100 = 132.25$. Confirmed. ::: ### Exercise 6.4 ★★ — Laspeyres, Paasche, Fisher with four goods A family consumes apples, chicken, olive oil, and pasta. In 2019: prices (EUR) 2.00, 5.50, 4.00, 1.20 and quantities 50, 30, 20, 40. In 2024: prices 2.80, 7.20, 8.50, 1.50 and quantities 40, 28, 12, 55. (a) Compute $L_P^{2024/2019}$, $P_P^{2024/2019}$, $F_P^{2024/2019}$. (b) Compute the value index $V^{2024/2019}$. (c) Verify the decomposition $V = L_P \cdot P_Q / 100$. (d) Identify the substitution effect from the data. ### Exercise 6.5 ★★ — Linking Spanish CPI across a base change Spain changed CPI base from 2016 to 2021. Available data: base 2016 reads 103.2 (2018), 104.0 (2019), 103.5 (2020), 106.8 (2021); base 2021 reads 100.0 (2021), 108.4 (2022), 112.1 (2023). (a) Express all values on base 2016 = 100. (b) Express all values on base 2021 = 100. (c) Compute cumulative inflation from 2018 to 2023. ### Exercise 6.6 ★★ — Deflating a nominal wage series Annual nominal salary (EUR) is 24,000, 24,600, 25,400, 26,800, 28,000 over 2019–2023. CPI (base 2019 = 100) is 100.0, 100.8, 103.5, 112.0, 116.2. (a) Compute the real salary (in 2019 EUR) for each year. (b) Compute cumulative *nominal* growth 2019 → 2023. (c) Compute cumulative *real* growth 2019 → 2023. (d) Comment on the worker's purchasing-power trajectory. ### Exercise 6.7 ★★ — Required salary and purchasing power A Granada professor earned EUR 36,000 in 2019. Spanish CPI (base 2016 = 100) is 104.5 (2019) and 121.3 (2024). (a) Compute the salary in 2024 that would preserve 2019 purchasing power. (b) If the actual 2024 salary is EUR 39,500, has purchasing power risen or fallen? (c) Compute the percentage change in purchasing power. ### Exercise 6.8 ★★★ — CPI aggregation and inflation drivers A four-sub-group CPI has weights and indices in 2022 (= 100 across the board), 2023, and 2024: | Sub-group | Weight | CPI 2023 | CPI 2024 | |-----------|-------:|---------:|---------:| | Food | 0.30 | 110.5 | 114.2 | | Housing | 0.35 | 104.0 | 108.6 | | Transport | 0.20 | 115.8 | 118.3 | | Other | 0.15 | 102.5 | 105.0 | (a) Compute the overall CPI for 2023 and 2024. (b) Compute year-on-year inflation rates for 2023 and 2024. (c) Compute each sub-group's contribution to 2023 inflation and identify the principal driver. (d) Has inflation accelerated or decelerated from 2023 to 2024? Comment in light of the substitution-bias warning in §6.4.3.