Comparing axiom systems

library(pgt)

Which axiom system for bad outputs?

How to model a technology that produces both a good and a bad output is an open question in the productivity literature. The 2021 symposium in the Journal of Productivity Analysis, built around a target article by Førsund (2021), which argues for a multi-equation model in the tradition of Frisch, with comments by Murty and Russell (2021), Ang and Dakpo (2021) and Färe and Grosskopf (2021) and Førsund’s (2021) rejoinder, set the weak-disposability school against the by-production school, with the materials-balance view running through both. The choice of axiom changes the efficiency scores, so a result that holds under one system need not hold under another. pgt implements the competing systems on one data object and reports how far their conclusions agree.

The models split into two families. The materials-balance models are built on the account \(u'x_l - v y_l \ge b_l\), which enters each programme differently:

  • wgd: weak-G-disposability (Hampf and Rødseth 2015, in the form of Rødseth 2025), where the account enters only as a valuation of retained output content in the objective.
  • mb_cost: the materials-balance cost model (Coelli et al. 2007), where the flow coefficients act as a price system on the inputs, reporting the technical and environmental-allocative components of \(EE = TE \times EAE\).

The reference models come from the competing systems:

  • byprod: the by-production intersection technology (Murty, Russell and Levkoff 2012), reporting output efficiency, emission efficiency and their average, the by-production efficiency index.
  • wd: weak disposability (Kuosmanen 2005), which imposes no materials balance.

Fitting each model

The examples use the synthetic steeldemo panel shipped with the package.

data(steeldemo)
tech <- pgt_tech(
  x = steeldemo[, c("coal_coke", "other_fuel", "raw_material", "flux",
                    "capture_energy")],
  y = steeldemo$production,
  b = steeldemo$emissions,
  a = steeldemo$captured,
  v = 0.01467,
  x_abate = "capture_energy",
  group = steeldemo$route,
  id = steeldemo$plant
)

fits <- lapply(c("wgd", "byprod", "mb_cost", "wd"), function(m) {
  pgt(tech, model = m)
})
names(fits) <- c("wgd", "byprod", "mb_cost", "wd")
sapply(fits, function(f) round(median(f$results$efficiency, na.rm = TRUE), 3))
#>     wgd  byprod mb_cost      wd 
#>   0.440   0.439   0.507   0.536

The by-production and materials-balance measures need not agree, because they reduce a different quantity: byprod contracts the observed emission over the emission-causing inputs, while mb_cost minimises the total material inflow.

The principal scores are all normalised so that 1 is efficient, but they are not the same quantity: wgd, byprod and wd report the emission ratio \(b^*/b\), while mb_cost reports the material-inflow ratio \(EE\). Only the rank-based statistics, the Spearman matrix and the quartile overlap below, are strictly comparable across models; the median column should be read model by model.

The comparison harness

compare_models() fits the efficiency-scored models and reports rank agreement and the overlap of their worst-performer sets.

cmp <- compare_models(tech, models = c("wgd", "byprod", "mb_cost", "wd"))
cmp
#> pgt model comparison: 4 models, returns = vrs, peers = all
#>   models: wgd, byprod, mb_cost, wd
#> 
#> Principal environmental efficiency by model (b*/b; EE for mb_cost):
#>    model n_solved median bottom_q_overlap
#>      wgd      180 0.4402           1.0000
#>   byprod      180 0.4385           0.6667
#>  mb_cost      180 0.5067           0.8667
#>       wd      180 0.5356           0.9556
#> 
#> Spearman rank correlation:
#>           wgd byprod mb_cost    wd
#> wgd     1.000  0.875   0.852 0.893
#> byprod  0.875  1.000   0.655 0.694
#> mb_cost 0.852  0.655   1.000 0.897
#> wd      0.893  0.694   0.897 1.000
#> 
#>   largest ranking disagreement: mb_cost vs byprod (rho = 0.655)

The Spearman matrix shows how closely the models rank the plants. A high correlation means the choice of axiom moves the scores but preserves the ordering; a low correlation warns that the ranking itself depends on the axiom. The bottom_q_overlap column reports, for each model, the share of its own worst-quartile plants that the reference model (the first model listed) also places in its worst quartile, the plants a regulator would target first.

plot(cmp)

Line plot of efficiency profiles under four models, with plants ordered by the first model's score and one line per model; where lines cross, the models disagree on a plant's rank

Reading the plot: plants are ordered along the horizontal axis by their weak-G-disposability score, and each line is one model. Where the lines track together the axiom choice is immaterial; where they cross, a plant’s relative standing depends on which system is used.

References

Ang, F., & Dakpo, K. H. (2021). Comment: Performance measurement and joint production of intended and unintended outputs. Journal of Productivity Analysis, 55(3), 185-188. doi:10.1007/s11123-021-00606-z

Coelli, T., Lauwers, L., & Van Huylenbroeck, G. (2007). Environmental efficiency measurement and the materials balance condition. Journal of Productivity Analysis, 28(1-2), 3-12. doi:10.1007/s11123-007-0052-8

Färe, R., & Grosskopf, S. (2021). Comments: Performance measurement and joint production of intended and unintended outputs. Journal of Productivity Analysis, 55(3), 189-193. doi:10.1007/s11123-021-00604-1

Førsund, F. R. (2021). Performance measurement and joint production of intended and unintended outputs. Journal of Productivity Analysis, 55(3), 157-175. doi:10.1007/s11123-021-00599-9

Førsund, F. R. (2021). Rejoinders to the comments on my paper “Performance measurement and joint production of intended and unintended outputs”. Journal of Productivity Analysis, 55(3), 195-201. doi:10.1007/s11123-021-00605-0

Hampf, B., & Rødseth, K. L. (2015). Carbon dioxide emission standards for U.S. power plants: An efficiency analysis perspective. Energy Economics, 50, 140-153. doi:10.1016/j.eneco.2015.04.001

Kuosmanen, T. (2005). Weak disposability in nonparametric production analysis with undesirable outputs. American Journal of Agricultural Economics, 87(4), 1077-1082. doi:10.1111/j.1467-8276.2005.00788.x

Murty, S., & Russell, R. R. (2021). A commentary on “Performance measurement and joint production of intended and unintended outputs” by Finn Førsund. Journal of Productivity Analysis, 55(3), 177-184. doi:10.1007/s11123-021-00603-2

Murty, S., Russell, R. R., & Levkoff, S. B. (2012). On modeling pollution-generating technologies. Journal of Environmental Economics and Management, 64(1), 117-135. doi:10.1016/j.jeem.2012.02.005

Rødseth, K. L. (2025). On the development of a unified, nonparametric materials balance-based efficiency analysis model and its applications. Journal of Productivity Analysis, 64(3), 305-319. doi:10.1007/s11123-025-00768-0