price_indexes: Price indexes
In gpindex: Generalized Price and Quantity Indexes

price_indexes

R Documentation

Price indexes

Description

Calculate a variety of price indexes using information on prices and quantities at two points in time.

Usage

arithmetic_index(type)

geometric_index(type)

harmonic_index(type)

laspeyres_index(p1, p0, q0, na.rm = FALSE)

paasche_index(p1, p0, q1, na.rm = FALSE)

jevons_index(p1, p0, na.rm = FALSE)

lowe_index(p1, p0, qb, na.rm = FALSE)

young_index(p1, p0, pb, qb, na.rm = FALSE)

fisher_index(p1, p0, q1, q0, na.rm = FALSE)

hlp_index(p1, p0, q1, q0, na.rm = FALSE)

lm_index(elasticity)

cswd_index(p1, p0, na.rm = FALSE)

cswdb_index(p1, p0, q1, q0, na.rm = FALSE)

bw_index(p1, p0, na.rm = FALSE)

stuvel_index(a, b)

arithmetic_agmean_index(elasticity)

geometric_agmean_index(elasticity)

lehr_index(p1, p0, q1, q0, na.rm = FALSE)

Arguments

`type`	The name of the index. See details for the possible types of indexes.
`p1`	Current-period prices.
`p0`	Base-period prices.
`q0`	Base-period quantities.
`na.rm`	Should missing values be removed? By default missing values for prices or quantities return a missing value.
`q1`	Current-period quantities.
`qb`	Period-b quantities for the Lowe/Young index.
`pb`	Period-b prices for the Lowe/Young index.
`elasticity`	The elasticity of substitution for the Lloyd-Moulton and AG mean indexes.
`a, b`	Parameters for the generalized Stuvel index.

Details

The arithmetic_index(), geometric_index(), and harmonic_index() functions return a function to calculate a given type of arithmetic, geometric (logarithmic), and harmonic index. Together, these functions produce functions to calculate the following indexes.

Arithmetic indexes
Carli
Dutot
Laspeyres
Palgrave
Unnamed index (arithmetic mean of Laspeyres and Palgrave)
Drobisch (arithmetic mean of Laspeyres and Paasche)
Walsh-I (arithmetic Walsh)
Marshall-Edgeworth
Geary-Khamis
Lowe
Young
Geometric indexes
Jevons
Geometric Laspeyres
Geometric Paasche
Geometric Young
Törnqvist (or Törnqvist-Theil)
Montgomery-Vartia / Vartia-I
Sato-Vartia / Vartia-II
Walsh-II (geometric Walsh)
Theil
Rao
Harmonic indexes
Coggeshall (equally weighted harmonic index)
Paasche
Harmonic Laspeyres
Harmonic Young

Along with the lm_index() function to calculate the Lloyd-Moulton index, these are just convenient wrappers for generalized_mean() and index_weights().

The Laspeyres, Paasche, Jevons, Lowe, and Young indexes are among the most common price indexes, and so they get their own functions. The laspeyres_index(), lowe_index(), and young_index() functions correspond to setting the appropriate type in arithmetic_index(); paasche_index() and jevons_index() instead come from the harmonic_index() and geometric_index() functions.

In addition to these indexes, there are also functions for calculating a variety of indexes not based on generalized means. The Fisher index is the geometric mean of the arithmetic Laspeyres and Paasche indexes; the Harmonic Laspeyres Paasche index is the harmonic analog of the Fisher index (8054 on Fisher's list). The Carruthers-Sellwood-Ward-Dalen and Carruthers-Sellwood-Ward-Dalen-Balk indexes are sample analogs of the Fisher index; the Balk-Walsh index is the sample analog of the Walsh index. The AG mean index is the arithmetic or geometric mean of the geometric and arithmetic Laspeyres indexes, weighted by the elasticity of substitution. The stuvel_index() function returns a function to calculate a Stuvel index of the given parameters. The Lehr index is an alternative to the Geary-Khamis index, and is the implicit price index for Fisher's index 4153.

Value

arithmetic_index(), geometric_index(), harmonic_index(), and stuvel_index() each return a function to compute the relevant price indexes; lm_index(), arithmetic_agmean_index(), and geometric_agmean_index() each return a function to calculate the relevant index for a given elasticity of substitution. The others return a numeric value giving the change in price between the base period and current period.

Note

There are different ways to deal with missing values in a price index, and care should be taken when relying on these functions to remove missing values. Setting na.rm = TRUE removes price relatives with missing information, either because of a missing price or a missing weight, while using all available non-missing information to make the weights.

Certain properties of an index-number formula may not work as expected when removing missing values if there is ambiguity about how to remove missing values from the weights (as in, e.g., a Törnqvist or Sato-Vartia index). The balanced() operator may be helpful, as it balances the removal of missing values across prices and quantities prior to making the weights.

References

Balk, B. M. (2008). Price and Quantity Index Numbers. Cambridge University Press.

Fisher, I. (1922). The Making of Index Numbers. Houghton Mifflin Company.

ILO, IMF, OECD, Eurostat, UN, and World Bank. (2020). Consumer Price Index Manual: Theory and Practice. International Monetary Fund.

von der Lippe, P. (2001). Chain Indices: A Study in Price Index Theory, Spectrum of Federal Statistics vol. 16. Federal Statistical Office, Wiesbaden.

von der Lippe, P. (2015). Generalized Statistical Means and New Price Index Formulas, Notes on some unexplored index formulas, their interpretations and generalizations. Munich Personal RePEc Archive paper no. 64952.

Selvanathan, E. A. and Rao, D. S. P. (1994). Index Numbers: A Stochastic Approach. MacMillan.

Examples

p0 <- price6[[2]]
p1 <- price6[[3]]
q0 <- quantity6[[2]]
q1 <- quantity6[[3]]
pb <- price6[[1]]
qb <- quantity6[[1]]

#---- Calculating price indexes ----

# Most indexes can be calculated by combining the appropriate weights
# with the correct type of mean

geometric_index("Laspeyres")(p1, p0, q0)
geometric_mean(p1 / p0, index_weights("Laspeyres")(p0, q0))

# Arithmetic Laspeyres index

laspeyres_index(p1, p0, q0)
arithmetic_mean(p1 / p0, index_weights("Laspeyres")(p0, q0))

# Harmonic calculation for the arithmetic Laspeyres

harmonic_mean(p1 / p0, index_weights("HybridLaspeyres")(p1, q0))

# Same as transmuting the weights

all.equal(
  scale_weights(index_weights("HybridLaspeyres")(p1, q0)),
  scale_weights(
    transmute_weights(1, -1)(p1 / p0, index_weights("Laspeyres")(p0, q0))
  )
)

# This strategy can be used to make more exotic indexes, like the
# quadratic-mean index (von der Lippe, 2001, p. 71)

generalized_mean(2)(p1 / p0, index_weights("Laspeyres")(p0, q0))

# Or the exponential mean index (p. 64)

log(arithmetic_mean(exp(p1 / p0), index_weights("Laspeyres")(p0, q0)))

# Or the arithmetic hybrid index (von der Lippe, 2015, p. 5)

arithmetic_mean(p1 / p0, index_weights("HybridLaspeyres")(p1, q0))
contraharmonic_mean(p1 / p0, index_weights("Laspeyres")(p0, q0))

# Unlike its arithmetic counterpart, the geometric Laspeyres can
# increase when base-period prices increase if some of these prices
# are small

gl <- geometric_index("Laspeyres")
p0_small <- replace(p0, 1, p0[1] / 5)
p0_dx <- replace(p0_small, 1, p0_small[1] + 0.01)
gl(p1, p0_small, q0) < gl(p1, p0_dx, q0)

#---- Price updating the weights in a price index ----

# Chain an index by price updating the weights

p2 <- price6[[4]]
laspeyres_index(p2, p0, q0)

I1 <- laspeyres_index(p1, p0, q0)
w_pu <- update_weights(p1 / p0, index_weights("Laspeyres")(p0, q0))
I2 <- arithmetic_mean(p2 / p1, w_pu)
I1 * I2

# Works for other types of indexes, too

harmonic_index("Laspeyres")(p2, p0, q0)

I1 <- harmonic_index("Laspeyres")(p1, p0, q0)
w_pu <- factor_weights(-1)(p1 / p0, index_weights("Laspeyres")(p0, q0))
I2 <- harmonic_mean(p2 / p1, w_pu)
I1 * I2

#---- Percent-change contributions ----

# Percent-change contributions for the Tornqvist index

w <- index_weights("Tornqvist")(p1, p0, q1, q0)
(con <- geometric_contributions(p1 / p0, w))

all.equal(sum(con), geometric_index("Tornqvist")(p1, p0, q1, q0) - 1)

#---- Missing values ----

# NAs get special treatment

p_na <- replace(p0, 6, NA)

# Drops the last price relative

laspeyres_index(p1, p_na, q0, na.rm = TRUE)

# Only drops the last period-0 price

sum(p1 * q0, na.rm = TRUE) / sum(p_na * q0, na.rm = TRUE)

#---- von Bortkiewicz decomposition ----

paasche_index(p1, p0, q1) / laspeyres_index(p1, p0, q0) - 1

wl <- scale_weights(index_weights("Laspeyres")(p0, q0))
pl <- laspeyres_index(p1, p0, q0)
ql <- quantity_index(laspeyres_index)(q1, q0, p0)

sum(wl * (p1 / p0 / pl - 1) * (q1 / q0 / ql - 1))

# Similar decomposition for geometric Laspeyres/Paasche

wp <- scale_weights(index_weights("Paasche")(p1, q1))
gl <- geometric_index("Laspeyres")(p1, p0, q0)
gp <- geometric_index("Paasche")(p1, p0, q1)

log(gp / gl)

sum(scale_weights(wl) * (wp / wl - 1) * log(p1 / p0 / gl))

#---- Consistency in aggregation ----

p0a <- p0[1:3]
p0b <- p0[4:6]
p1a <- p1[1:3]
p1b <- p1[4:6]
q0a <- q0[1:3]
q0b <- q0[4:6]
q1a <- q1[1:3]
q1b <- q1[4:6]

# Indexes based on the generalized mean with value share weights are
# consistent in aggregation

lm_index(0.75)(p1, p0, q0)

w <- index_weights("LloydMoulton")(p0, q0)
Ia <- generalized_mean(0.25)(p1a / p0a, w[1:3])
Ib <- generalized_mean(0.25)(p1b / p0b, w[4:6])
generalized_mean(0.25)(c(Ia, Ib), c(sum(w[1:3]), sum(w[4:6])))

# Agrees with group-wise indexes

all.equal(lm_index(0.75)(p1a, p0a, q0a), Ia)
all.equal(lm_index(0.75)(p1b, p0b, q0b), Ib)

# Care is needed with more complex weights, e.g., Drobisch, as this
# doesn't fit Balk's (2008) definition (p. 113) of a generalized-mean
# index (it's the arithmetic mean of a Laspeyres and Paasche index)

arithmetic_index("Drobisch")(p1, p0, q1, q0)

w <- index_weights("Drobisch")(p1, p0, q1, q0)
Ia <- arithmetic_mean(p1a / p0a, w[1:3])
Ib <- arithmetic_mean(p1b / p0b, w[4:6])
arithmetic_mean(c(Ia, Ib), c(sum(w[1:3]), sum(w[4:6])))

# Does not agree with group-wise indexes

all.equal(arithmetic_index("Drobisch")(p1a, p0a, q1a, q0a), Ia)
all.equal(arithmetic_index("Drobisch")(p1b, p0b, q1b, q0b), Ib)

gpindex documentation built on Nov. 15, 2023, 9:06 a.m.