Index-number formulas

Price indexes based on a generalized mean of price relatives constitute a large family of bilateral index-number formulas that are consistent in aggregation, and any index based on the generalized mean can be used to make and aggregate elementary indexes. See Section 4 for a comprehensive list. To see how to make superlative indexes as well, let’s start with a simple dataset of prices and quantities for two businesses over three periods. We’ll be making extensive use of the lower-level math functions in this package functions; see the help pages for these functions to get more detail and vignette("decomposing-indexes") for the theoretical background.

library(piar)

prices <- data.frame(
  period = rep(1:3, each = 6),
  product = paste0("P", 1:6),
  business = rep(c("B1", "B2"), each = 3),
  price = 1:18,
  quantity = 18:1
)

prices[c("back_price", "back_quantity")] <-
  prices[back_period(prices$period, prices$product), c("price", "quantity")]

head(prices)
#>   period product business price quantity back_price back_quantity
#> 1      1      P1       B1     1       18          1            18
#> 2      1      P2       B1     2       17          2            17
#> 3      1      P3       B1     3       16          3            16
#> 4      1      P4       B2     4       15          4            15
#> 5      1      P5       B2     5       14          5            14
#> 6      1      P6       B2     6       13          6            13

Basic indexes

As seen in vignette("piar"), by default the elementary_index() function calculates a Jevons index (equally-weighted geometric mean of price relatives). Although this is the standard index-number formula for making elementary indexes, many other types of index-numbers are possible. Among the unweighted index-number formulas, the Carli index (equally-weighted arithmetic mean of price relatives) is the historical competitor to the Jevons, and requires specifying the order of the generalized mean r when calling elementary_index(). An order of 1 corresponds to an arithmetic mean.

prices |>
  elementary_index(price / back_price ~ period + business, r = 1)
#> Period-over-period price index for 2 levels over 3 time periods 
#>       time
#> levels 1        2        3
#>     B1 1 4.666667 1.757937
#>     B2 1 2.233333 1.548485

The Coggeshall index (equally-weighted harmonic mean of price relatives) is another competitor to the Jevons, but is seldom used in practice. Despite it being more exotic, it is just as easy to make by specifying an order r of -1.

prices |>
  elementary_index(price / back_price ~ period + business, r = -1)
#> Period-over-period price index for 2 levels over 3 time periods 
#>       time
#> levels 1        2        3
#>     B1 1 4.131148 1.754499
#>     B2 1 2.214765 1.547408

Weights can be added to make, for example, elementary indexes using the geometric Laspeyres formula.

prices |>
  elementary_index(
    price / back_price ~ period + business,
    weights = back_price * back_quantity
  )
#> Period-over-period price index for 2 levels over 3 time periods 
#>       time
#> levels 1        2        3
#>     B1 1 3.853330 1.754015
#>     B2 1 2.202496 1.549081

The type of mean used to aggregate elementary indexes can be controlled in the same way in the call to aggregate(). The default makes an arithmetic index, but any type of generalized-mean index is possible.

Superlative indexes

Many superlative indexes can be made by supplying unequal and time-varying weights when making the elementary indexes, usually from information about quantities. The Törnqvist index is a popular superlative index-number formula, using average period-over-period value shares as the weights in a geometric mean. As elementary_index() makes a geometric index by default, all that is needed to make a Törnqvist index is the weights.

period_by_business <- interaction(prices$period, prices$business)

tornqvist_weights <- split(prices, ~period_by_business) |>
  lapply(\(df) {
    0.5 * scale_weights(df$price * df$quantity) +
      0.5 * scale_weights(df$back_price * df$back_quantity)
  }
  ) |>
  unsplit(period_by_business)

prices |>
  elementary_index(
    price / back_price ~ period + business,
    weights = tornqvist_weights
  )
#> Period-over-period price index for 2 levels over 3 time periods 
#>       time
#> levels 1        2        3
#>     B1 1 4.087422 1.759213
#>     B2 1 2.215995 1.556014

Making a Fisher index is more complex because it does not belong to the generalized-mean family. Despite this, it is possible to make weights to represent a Fisher index as a generalized mean of any order.

fisher_weights <- split(prices, ~period_by_business) |>
  lapply(\(df) {
    transmute_weights2(
      df$price / df$back_price,
      list(df$back_price * df$back_quantity, df$price * df$quantity),
      to = 0
    )
  }) |>
  unsplit(period_by_business)

prices |>
  elementary_index(
    price / back_price ~ period + business,
    weights = fisher_weights
  )
#> Period-over-period price index for 2 levels over 3 time periods 
#>       time
#> levels 1        2        3
#>     B1 1 4.076840 1.759215
#>     B2 1 2.215934 1.556046

Aggregating with a superlative index is more complex, and is the subject of vignette("superlative-aggregation").

Product contributions

As shown in vignette("contributions"), supplying contrib = TRUE in the call to elementary_index() makes percent-change product contributions for each index value. The method used in this package is flexible and works well for a variety of different index-number formulas, but does not include all methods found in the literature. (See the help page for elementary_index(), and the references therein, for more detail about the exact methods.)

Let’s extend the previous example by calculating product contributions for the Fisher index using the default method.

fisher_index <- prices |>
  elementary_index(
    price / back_price ~ period + business,
    weights = fisher_weights,
    contrib = TRUE
  )

contrib(fisher_index, "B1")
#>        time
#> product 1         2         3
#>    B1.1 0 1.1024526 0.2899319
#>    B1.2 0 1.0256151 0.2530718
#>    B1.3 0 0.9487724 0.2162114

We can change the method used to make contributions for, say, business B1 by making a function to compute contributions according to a different method

diewert_contributions <- function(p1, p0, q1, q0) {
  rel <- p1 / p0
  v0 <- scale_weights(p0 * q0)
  v1 <- scale_weights(p0 * q1)
  laspeyres <- gmean(rel, v0)
  fisher <- nested_gmean(rel, list(v0, v1), order = c(1, 1))

  (v0 + laspeyres * v1) / (1 + fisher) * (rel - 1)
}

and using this function to replace the contributions for business B1.

contrib(fisher_index, "B1") <- subset(prices, business == "B1") |>
  split(~period) |>
  sapply(
    \(df) {
      diewert_contributions(
        df$price,
        df$back_price,
        df$quantity,
        df$back_quantity
      )
    }
  )

contrib(fisher_index, "B1")
#>        time
#> product 1         2         3
#>       1 0 1.1124046 0.2937263
#>       2 0 1.0256134 0.2530717
#>       3 0 0.9388222 0.2124170

Aggregating the elementary indexes will then consistently aggregate the contributions for both businesses, even though they use different methods.

Indexes based on generalized means

# ---- Arithmetic mean ----
# Carli
carli <- \(p1, p0, q1, q0) gmean(p1 / p0)

# Dutot
dutot <- \(p1, p0, q1, q0) gmean(p1 / p0, p0)

# Laspeyres
laspeyres <- \(p1, p0, q1, q0) gmean(p1 / p0, p0 * q0)

# Palgrave
palgrave <- \(p1, p0, q1, q0) gmean(p1 / p0, p1 * q1)

# Walsh-1
walsh1 <- \(p1, p0, q1, q0) gmean(p1 / p0, p0 * sqrt(q0 * q1))

# Marshall-Edgeworth
marshall_edgeworth <- \(p1, p0, q1, q0) gmean(p1 / p0, p0 * (q0 * q1))

# Geay-Khamis
geary_khamis <- \(p1, p0, q1, q0) gmean(p1 / p0, p0 / (1 / q0 + 1 / q1))

# Hybrid CSWD
hybrid_cswd <- \(p1, p0, q1, q0) gmean(p1 / p0, sqrt(p0 / p1))

# Martini
martini <- \(p1, p0, q1, q0, a) gmean(p1 / p0, p0 * q0 * (q1 / p0)^a)

# ---- Geometric mean ----
# Jevons
jevons <- \(p1, p0, q1, q0) gmean(p1 / p0, order = 0)

# Geometric Laspeyres (Jöhr)
geo_laspeyres <- \(p1, p0, q1, q0) gmean(p1 / p0, p0 * q0, order = 0)

# Geometric Paasche
geo_paasche <- \(p1, p0, q1, q0) gmean(p1 / p0, p1 * q1, order = 0)

# Walsh-2
walsh2 <- \(p1, p0, q1, q0) gmean(p1 / p0, sqrt(p0 * p1 * q0 * q1), order = 0)

# Theil
theil <- \(p1, p0, q1, q0) {
  w0 <- scale_weights(p0 * q0)
  w1 <- scale_weights(p1 * q1)
  gmean(p1 / p0, ((w0 + w1) / 2 * w0 * w1)^(1 / 3), order = 0)
}

# Rao
rao <- \(p1, p0, q1, q0) {
  w0 <- scale_weights(p0 * q0)
  w1 <- scale_weights(p1 * q1)
  gmean(p1 / p0, w0 * w1 / (w0 + w1), order = 0)
}

# Sato-Vartia
sato_vartia <- function(p1, p0, q1, q0) {
  v0 <- scale_weights(p0 * q0)
  v1 <- scale_weights(p1 * q1)
  gmean(p1 / p0, emean(v0, v1), order = 0)
}

# ---- Harmonic mean ----
# Coggeshall
coggeshall <- \(p1, p0, q1, q0) gmean(p1 / p0, order = -1)

# Paasche
paasche <- \(p1, p0, q1, q0) gmean(p1 / p0, p1 * q1, order = -1)

# Harmonic Laspeyres
harm_laspeyres <- \(p1, p0, q1, q0) gmean(p1 / p0, p0 * q0, order = -1)

# ---- Generalized mean ----
# Lloyd-Moulton
lloyd_moulton <- \(p1, p0, q1, q0, sigma) {
  gmean(p1 / p0, p0 * q0, order = 1 - sigma)
}

# ---- Nested means ----
# Drobisch (Sidgwick)
drobisch <- \(p1, p0, q1, q0) {
  nested_gmean(p1 / p0, list(p0 * q0, p1 * q1), outer_order = 1)
}

# Unnamed
unnamed <- \(p1, p0, q1, q0) {
  nested_gmean(
    p1 / p0,
    list(p0 * q0, p1 * q1),
    order = c(1, 1),
    outer_order = 1
  )
}

# Törnqvist-Theil
tornqvist <- \(p1, p0, q1, q0) {
  nested_gmean(
    p1 / p0,
    list(p0 * q0, p1 * q1),
    order = c(0, 0)
  )
}

# Fisher
fisher <- \(p1, p0, q1, q0) nested_gmean(p1 / p0, list(p0 * q0, p1 * q1))

# CSWD
cswd <- \(p1, p0, q1, q0) nested_gmean(p1 / p0)

# Balk-Walsh
balk_walsh <- \(p1, p0, q1, q0) nested_gmean(p1 / p0, order = c(0.5, -0.5))

# Geometric AG mean
ag_mean <- \(p1, p0, q1, q0, elasticity) {
  nested_gmean(
    p1 / p0,
    list(p0 * q0, p0 * q0),
    order = c(0, 1),
    outer_weights = c(elasticity, 1 - elasticity)
  )
}