contributions: Percent-change contributions

In marberts/gpindex: Generalized Price and Quantity Indexes

Percent-change contributions

Description

Calculate additive percent-change contributions for generalized-mean price indexes, and indexes that nest two levels of generalized means consisting of an outer generalized mean and two inner generalized means (e.g., the Fisher index).

Usage

contributions(r)

arithmetic_contributions(x, w = NULL)

geometric_contributions(x, w = NULL)

harmonic_contributions(x, w = NULL)

nested_contributions(r1, r2, t = c(1, 1))

nested_contributions2(r1, r2, t = c(1, 1))

fisher_contributions(x, w1 = NULL, w2 = NULL)

fisher_contributions2(x, w1 = NULL, w2 = NULL)

Arguments

r	A finite number giving the order of the generalized mean.
x	A strictly positive numeric vector.
w, w1, w2	A strictly positive numeric vector of weights, the same length as x. The default is to equally weight each element of x.
r1	A finite number giving the order of the outer generalized mean.
r2	A pair of finite numbers giving the order of the inner generalized means.
t	A pair of strictly positive weights for the inner generalized means. The default is equal weights.

Details

The function contributions() is a simple wrapper for transmute_weights(r, 1)() to calculate (additive) percent-change contributions for a price index based on a generalized mean of order r. It returns a function to compute a vector v(x, w) such that

generalized_mean(r)(x, w) - 1 == sum(v(x, w))

The arithmetic_contributions(), geometric_contributions() and harmonic_contributions() functions cover the most important cases (i.e., r = 1, r = 0, and r = -1).

The nested_contributions() and nested_contributions2() functions are the analog of contributions() for an index based on a nested generalized mean with two levels, like a Fisher index. They are wrappers around nested_transmute() and nested_transmute2(), respectively.

The fisher_contributions() and fisher_contributions2() functions correspond to nested_contributions(0, c(1, -1))() and nested_contributions2(0, c(1, -1))(), and are appropriate for calculating percent-change contributions for a Fisher index.

Value

contributions() returns a function:

function(x, w = NULL){...}

nested_contributions() and nested_contributions2() return a function:

function(x, w1 = NULL, w2 = NULL){...}

arithmetic_contributions(), geometric_contributions(), harmonic_contributions(), fisher_contributions(), and fisher_contributions2() each return a numeric vector, the same length as x.

References

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

Hallerbach, W. G. (2005). An alternative decomposition of the Fisher index. Economics Letters, 86(2):147–152

Reinsdorf, M. B., Diewert, W. E., and Ehemann, C. (2002). Additive decompositions for Fisher, Törnqvist and geometric mean indexes. Journal of Economic and Social Measurement, 28(1-2):51–61.

See Also

transmute_weights() for the underlying implementation.

Examples

p2 <- price6[[2]]
p1 <- price6[[1]]
q2 <- quantity6[[2]]
q1 <- quantity6[[1]]

# Percent-change contributions for the Jevons index.

geometric_mean(p2 / p1) - 1

geometric_contributions(p2 / p1)

all.equal(
  geometric_mean(p2 / p1) - 1,
  sum(geometric_contributions(p2 / p1))
)

# Percent-change contributions for the Fisher index in section 6 of
# Reinsdorf et al. (2002).

(con <- fisher_contributions(p2 / p1, p1 * q1, p2 * q2))

all.equal(sum(con), fisher_index(p2, p1, q2, q1) - 1)

# Not the only way.

(con2 <- fisher_contributions2(p2 / p1, p1 * q1, p2 * q2))

all.equal(sum(con2), fisher_index(p2, p1, q2, q1) - 1)

# The same as the van IJzeren decomposition in section 4.2.2 of
# Balk (2008).

Qf <- quantity_index(fisher_index)(q2, q1, p2, p1)
Ql <- quantity_index(laspeyres_index)(q2, q1, p1)
wl <- scale_weights(p1 * q1)
wp <- scale_weights(p1 * q2)

(Qf / (Qf + Ql) * wl + Ql / (Qf + Ql) * wp) * (p2 / p1 - 1)

# Similar to the method in section 2 of Reinsdorf et al. (2002),
# although those contributions aren't based on weights that sum to 1.

Pf <- fisher_index(p2, p1, q2, q1)
Pl <- laspeyres_index(p2, p1, q1)

(1 / (1 + Pf) * wl + Pl / (1 + Pf) * wp) * (p2 / p1 - 1)

# Also similar to the decomposition by Hallerbach (2005), noting that
# the Euler weights are close to unity.

Pp <- paasche_index(p2, p1, q2)

(0.5 * sqrt(Pp / Pl) * wl + 0.5 * sqrt(Pl / Pp) * wp) * (p2 / p1 - 1)

marberts/gpindex documentation built on June 10, 2025, 5:06 a.m.