Overview with examples of common bilateral methods
Serge Goussev
September 27, 2024
March 11, 2026
A price index provides an aggregate measure of price change for a particular product segment, industry, or overall economy. … [They compare] the cost of purchasing a set of goods at different points in time. This “set of goods” is often referred to as the “market basket” or the “bundle” of goods. - Aizcorbe (2014)
To illustrate how each index can be calculated, I’ll illustrate using some example datasets and some common price index methods.
gpindex
gpindex is a convenient R package for price index calculation. It has a simple play dataset built in that we can use to demonstrate how to construct various price indices.
Warning: package 'gpindex' was built under R version 4.5.2 t1 t2 t3 t4 t5
1 1 1.2 1.0 0.8 1.0
2 1 3.0 1.0 0.5 1.0
3 1 1.3 1.5 1.6 1.6
4 1 0.7 0.5 0.3 0.1
5 1 1.4 1.7 1.9 2.0
6 1 0.8 0.6 0.4 0.2 t1 t2 t3 t4 t5
1 1.0 0.8 1.0 1.2 0.9
2 1.0 0.9 1.1 1.2 1.2
3 2.0 1.9 1.8 1.9 2.0
4 1.0 1.3 3.0 6.0 12.0
5 4.5 4.7 5.0 5.6 6.5
6 0.5 0.6 0.8 1.3 2.5The simplest formula (and one that is commonly used) is the Laspeyres index:
\[ I^{L}_{0,1} = \frac{\sum_{n=1}^{N}(p_n^1q_n^0)}{\sum_{n=1}^{N}(p_n^0q_n^0)} \]
Where \(p\) and \(q\) denote prices and quantities, and 0 and 1 denote two points in time.
If \(N\) goods are sold in both periods (note that an overlap is needed), we can compare the the cost of purchasing the same goods we bought in period 1 with a certain period in the future.
We can also write the Laspeyres as the weighted arithmetic average of the price change of the individual products in the index
\[ I^L_{0,1} = \sum_{n=1}^{N} (w_n^0\frac{p_n^1}{p_n^0}) \]
\[ w_m^0 = (p_n^0q_n^0)/\sum_{n=1}^{N}(p_n^0q_n^0) \]
give the ratio of good \(n\) expenditure to total expenditure, or could also be considered the relative importance or share of the product. There are some key nuances with this approach:
Similar to the Lasperyes, however uses a different basket (the one from the pricing period):
\[ I^P_{0,1} = \sum_{n=1}^N ( p_n^1 q_n^1 ) / \sum_{n=1}^N ( p_n^0 q_n^1 ) \]
The Paasche may also be expressed as a function of the weighted average (i.e. shares):
\[ I^P_{0,1} = 1/ \sum_{n=1}^N ( w_n^1 \frac {p_n^0} {p_n^1} ) \]
\[ w_{n,1} = ( p_n^1 q_n^1 )/ \sum_{n=1}^n ( w_n^1 p_n^0 q_n^1 ) \]
The Laspeyres and Paasche include prices and quantities in both periods, and both use the same relatives with different expenditure shares.
The Fisher does a geometric average of the Laspeyres and the Paasche:
\[ I^F_{0,1} = ( I^L_{0,1} I^P_{0,1})^\frac{1}{2} \]
The Fisher thus uses expenditure from both periods and thus provides relative importance that are more closely aligned with the goods actually sold. As the Fisher satisfies homogeneity, symmetry, and the time reversal test (the price change from the base to the current should be the inverse of the current to the base) - thus it doesn’t matter what period is chosen as the base.
Similar to the Fisher, however it takes the average of the weights instead of averaging the two indices
In logged form: \[ lnI^T_{0,1}= \sum^N_{n=1} ( w_{n,0}+ w_{n,1} )/2 ( ln\frac {p_{n,1}} {p_{n,0}} ) \]
We can also view the Törnqvist as an exponenet of logged form (quite similar to the above):
\[ P^T(p^1,q^1,p^0,q^0) = exp\{ \sum_{n=1}^N \frac{1}{2} (s_n^1+s_n^1) ln(p_n^1/p_n^0) \} \]
And also in more traditional form:
\[ P^T(p^1,q^1,p^0,q^0) = \prod_{n=1}^N \left( \frac{p_n^1}{p_n^0} \right) ^{\frac{s_n^1+s_n^0}{2}} \]
gpindex
As there is no out of the box function for the Törnqvist, we can use the geometric_mean() function.
The Jevons is unweighted geometric mean. Similar to the Törnqvist, in logged form:
\[ lnI^J_{0,1} = \frac{1}{N} \sum^N_{n=1} ln(p_n^1/p_n^0) \]
The Jevons takes the unweighted average by replacing the \((w_{m,0}+w_{m,1})/2\) with \(1/M\), thus giving each model equal weight
The Jevons can also be written in simpler form as per Balk(2008) (formula 1.5).
\[ P^J(p^1,p^0) = \prod_{n=1}^N \left( \frac{p_n^1}{p_n^0} \right) ^{1/N} \]
Using the full toy dataset - lets compare the price indices
# 1. Use t1 as the base period
p_base <- price6$t1
q_base <- quantity6$t1
# 2. Loop through the periods to calculate indices for the set
results <- mapply(function(p_curr, q_curr) {
# Calculate current value shares (s1) for Törnqvist weights
s1 <- (p_curr * q_curr) / sum(p_curr * q_curr)
s2 <- (p_base * q_base) / sum(p_base * q_base)
w_torn <- (s1 + s2) / 2
c(
Laspeyres = laspeyres_index(p_curr, p_base, q_base),
Paasche = paasche_index(p_curr, p_base, q_curr),
Fisher = fisher_index(p_curr, p_base, q_curr, q_base),
Tornqvist = geometric_mean(p_curr / p_base, w_torn),
Jevons = geometric_mean(p_curr / p_base)
)
}, price6, quantity6)
# 3. Convert to a clean dataframe
df_indexes <- as.data.frame(t(results))
df_indexes$Period <- rownames(df_indexes)
## Now lets plot this (for which we need a few additional libraries)
library(plotly)
library(ggplot2)
library(tidyr)
# 4. Create the plot
fig <- plot_ly(df_indexes, x = ~Period, width = 800, height = 500) %>%
add_trace(y = ~Laspeyres, name = 'Laspeyres', type = 'scatter', mode = 'lines+markers') %>%
add_trace(y = ~Paasche, name = 'Paasche', type = 'scatter', mode = 'lines+markers') %>%
add_trace(y = ~Fisher, name = 'Fisher', type = 'scatter', mode = 'lines+markers',
line = list(dash = 'dash')) %>%
add_trace(y = ~Tornqvist, name = 'Törnqvist', type = 'scatter', mode = 'lines+markers',
line = list(dash = 'dot')) %>%
add_trace(y = ~Jevons, name = 'Jevons', type = 'scatter', mode = 'lines+markers') %>%
layout(
title = "Interactive Comparison of Five Price Index Formulas",
yaxis = list(title = "Index Value (Base t1 = 1.0)"),
xaxis = list(title = "Time Period"),
hovermode = "x unified", # Show values in one tooltip when hovering
legend = list(orientation = 'h', y = -0.2) # Move legend to bottom
)
fig