gpindex (version 0.6.2)

price_indexes: 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)

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.

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 (or Sidgwick, arithmetic mean of Laspeyres and Paasche)

• Walsh-I (arithmetic Walsh)

• Marshall-Edgeworth

• Geary-Khamis

• Lowe

• Young

• Hybrid-CSWD

• Geometric indexes

• Jevons

• Geometric Laspeyres (or Jöhr)

• 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 (or Harmonic Paasche Laspeyres) 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.

References

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

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

IMF, ILO, Eurostat, UNECE, OECD, and World Bank. (2020). Consumer Price Index Manual: Concepts and Methods. International Monetary Fund.

von der Lippe, P. (2007). Index Theory and Price Statistics. Peter Lang.

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.

See Also

generalized_mean() for the generalized mean that powers most of these functions.

contributions() for calculating percent-change contributions.

quantity_index() to remap the arguments in these functions for a quantity index.

price6() for an example of how to use these functions with more than two time periods.

The piar package has more functionality working with price indexes for multiple groups of products over many time periods.

Other price index functions: geks(), index_weights(), splice_index()

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)),
  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, 2007, p. 61)

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

# Or the exponential mean index (p. 62)

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)