---
title: "Comparing axiom systems"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Comparing axiom systems}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 6,
                      fig.height = 4)
```

```{r setup}
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*, 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.

```{r}
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))
```

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.

```{r}
cmp <- compare_models(tech, models = c("wgd", "byprod", "mb_cost", "wd"))
cmp
```

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.

```{r, fig.alt = "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"}
plot(cmp)
```

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