Generalized Price and Quantity Indexes

Generalized-mean price indexes are a large family of price indexes with nice properties, such as the mean-value and identity properties (e.g., Balk, 2008, Chapter 3). When used with value-share weights, these indexes satisfy the key homogeneity properties, commensurability, and are consistent in aggregation. This last feature makes generalized-mean indexes natural candidates for making national statistics, and this justifies the hierarchical structure used by national statistical agencies for calculating and disseminating collections of price indexes.

Almost all bilateral price indexes used in practice are either generalized-mean indexes (like the Laspeyres and Paasche index) or are nested generalized-mean indexes (like the Fisher index). The purpose of this vignette is to show some of the key functions for working with generalized-mean indexes. In what follows, everything is framed as a price index to avoid duplication; it is trivial to turn a price index into its analogous quantity index by simply switching prices and quantities.

## Making generalized-mean indexes

A generalized-mean price index is a weighted generalized mean of price relatives. Given a set of price relatives and weights, any generalized-mean price index is easily calculated with the `generalized_mean()` function. What distinguishes different generalized-mean price indexes are the weights and the order of the generalized mean. For example, the standard Laspeyres index uses base-period value-share weights in a generalized mean of order 1 (arithmetic mean).

    library(gpindex)

    # Start with some data on prices and quantities for 6 products
    # over 5 periods
    price6
    #>   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
    quantity6
    #>    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  2.5

    # We'll only need prices and quantities for a few periods
    p1 <- price6[[1]]
    p2 <- price6[[2]]
    p2 <- price6[[3]]
    q1 <- quantity6[[1]]
    q2 <- quantity6[[2]]

    s1 <- p1 * q1
    s2 <- p1 * q2

    # Laspeyres index
    arithmetic_mean(p1 / p1, s1)
    #> [1] 1

Changing the order of the generalized mean to \(1 - \sigma\), where \(\sigma\) is an elasticity of substitution, gives a Lloyd-Moulton index, whereas changing the weights to current-period value-shares gives a Palgrave index. This is the essence of the atomistic approach in chapter 2 of Selvanathan and Rao (1994).

    # Lloyd-Moulton index (elasticity of substitution -1)
    quadratic_mean <- generalized_mean(2)
    quadratic_mean(p1 / p1, s1)
    #> [1] 1

    # Palgrave index
    arithmetic_mean(p1 / p1, s2)
    #> [1] 1

Generalized-mean indexes can also be nested together to get indexes like the Fisher, Drobisch, or AG mean index. The `nested_mean()` function is a simple wrapper for `generalized_mean()` for these cases.

    # Fisher index
    fisher_mean(p1 / p1, s1, s2)
    #> [1] 1

    # Drobisch index
    drobisch_mean <- nested_mean(1, c(1, 1))
    drobisch_mean(p1 / p1, s1, s2)
    #> [1] 1

    # Geometric AG mean index (elasticity of substitution 0.25)
    ag_mean <- nested_mean(0, c(0, 1), c(0.25, 0.75))
    ag_mean(p1 / p1, s1, s1)
    #> [1] 1

On top of these basic mathematical tools are functions for making standard price indexes when both prices and quantities are known. Weights for a large variety of indexes can be calculated with `index_weights()`, which can be plugged into the relevant generalized mean to calculate most common price indexes, and many uncommon ones. The `price_index` functions provide a simple wrapper, with the `quantity_index()` function turning each of these into its analogous quantity index.

    # Laspeyres index, again
    arithmetic_mean(p1 / p1, index_weights("Laspeyres")(p1, q1))
    #> [1] 1

    laspeyres_index(p1, p1, q1)
    #> [1] 1

## Decomposing indexes

Two important functions for decomposing generalized means are given by `transmute_weights()` and `factor_weights()`. These functions augment the weights in a generalized mean, and can be used to calculate percent-change contributions (with, e.g., `contributions()`) and price-update weights for generalized-mean indexes.

    quadratic_decomposition <- transmute_weights(2, 1)

    arithmetic_mean(p1 / p1, quadratic_decomposition(p1 / p1, s1))
    #> [1] 1
    quadratic_mean(p1 / p1, s1)
    #> [1] 1

    quadratic_contributions <- contributions(2)
    quadratic_contributions(p1 / p1, s1)
    #> [1] 0 0 0 0 0 0

    quadratic_update <- factor_weights(2)

    quadratic_mean(p2 / p1, s1)
    #> [1] 1.411914
    quadratic_mean(p2 / p1, quadratic_update(p1 / p1, s1)) *
      quadratic_mean(p1 / p1, s1)
    #> [1] 1.411914

Percent-change contributions can similarly be calculated for indexes that nest generalized means.

    ag_decomposition <- nested_transmute(0, c(0, 1), 1, c(0.25, 0.75))

    ag_mean(p1 / p1, s1, s1)
    #> [1] 1
    arithmetic_mean(p1/ p1, ag_decomposition(p1 / p1, s1, s1))
    #> [1] 1

    ag_contributions <- nested_contributions(0, c(0, 1), c(0.25, 0.75))
    ag_contributions(p1 / p1, s1, s1)
    #> [1] 0 0 0 0 0 0

## References

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

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