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, anchored by Førsund
(2021) and continued in the comments of Murty and Russell (2021), Ang
and Dakpo (2021) and Färe and Grosskopf (2021), with 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
enforce \(u'x_l - v y_l \ge
b_l\):
wgd: weak-G-disposability (Rødseth 2025).mb_cost: the materials-balance cost model (Coelli et
al. 2007), 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 graph average.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")],
y = steeldemo$production,
b = steeldemo$emissions,
v = 0.01467,
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
#> 0.485 0.981 0.509 0.475
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 headline 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
#>
#> Headline environmental efficiency by model (b*/b; EE for mb_cost):
#> model n_solved median bottom_q_overlap
#> wgd 180 0.4850 1.0000
#> byprod 180 0.9812 0.4222
#> mb_cost 180 0.5095 0.9556
#> wd 180 0.4752 1.0000
#>
#> Spearman rank correlation:
#> wgd byprod mb_cost wd
#> wgd 1.000 0.455 0.991 1.000
#> byprod 0.455 1.000 0.401 0.452
#> mb_cost 0.991 0.401 1.000 0.991
#> wd 1.000 0.452 0.991 1.000
#>
#> largest ranking disagreement: mb_cost vs byprod (rho = 0.401)
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.
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
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