The estimators with a numerical example printed in their source paper are validated against it: the weak-G-disposability and directional models against Rødseth (2025) Tables 2 and 3, and the by-production model against the analytic Example 1 of Murty, Russell and Levkoff (2012). These replications run as unit tests on every check; this vignette reproduces them in the open. The remaining estimators carry analytic hand-computed checks in the test suite rather than published-table replications; the final section shows the internal-consistency check for the materials-balance cost model.
The pigfarms data is Table 1 of Rødseth (2025): five
pig-finishing farms with feed and piglets as material inputs, meat as
the good output, and controlled, uncontrolled and abatement emission
columns for nitrogen. Because the paper does not print the nitrogen
coefficients of feed and piglets, the uncontrolled-emission aggregate
replaces them as the single material carrier (\(u = 1\), \(v =
0\)); the uncontrolled emission equals controlled plus abatement,
so the accounts close exactly. Keeping all five columns as inputs with
zero material coefficients on feed and piglets (the mapping in the
pigfarms help page) yields the same Table 2 minima; the
test suite asserts the two mappings agree.
data(pigfarms)
pigfarms
#> farm feed piglet labor capital meat controlled uncontrolled abatement
#> 1 A 21 2 50 100 10 16 16.7 0.7
#> 2 B 21 2 50 100 7 16 20.2 4.2
#> 3 C 21 2 50 100 7 20 20.2 0.2
#> 4 D 24 3 50 100 11 20 20.4 0.4
#> 5 E 24 3 50 100 10 21 21.6 0.6tech <- pgt_tech(
x = pigfarms[, c("uncontrolled", "labor", "capital")],
y = pigfarms$meat,
b = pigfarms$controlled,
u = c(1, 0, 0),
a = pigfarms$abatement,
id = pigfarms$farm
)
fit <- pgt(tech, model = "wgd")
data.frame(farm = fit$results$id, b_star = fit$results$b_star)
#> farm b_star
#> 1 A 16
#> 2 B 16
#> 3 C 16
#> 4 D 20
#> 5 E 16The minimal controlled emissions are \((16, 16, 16, 20, 16)\), which match the paper’s Table 2.
The factorially determined multi-output model (Eq. 13, abatement fixed at each farm’s own level) jointly expands the good output and contracts the bad output. Its good-output coefficient \(v = 7/6\) follows from farms A and B, which share inputs: their meat differs by 3 and their uncontrolled emissions by 3.5. Each farm’s feed coefficient closes its account to the uncontrolled column.
v <- 7 / 6
u_feed <- (pigfarms$uncontrolled + v * pigfarms$meat) / pigfarms$feed
tech3 <- pgt_tech(
x = pigfarms[, c("feed", "piglet", "labor", "capital")],
y = pigfarms$meat,
b = pigfarms$controlled,
u = cbind(u_feed, 0, 0, 0),
v = v,
a = pigfarms$abatement,
id = pigfarms$farm
)
fdmo <- pgt(tech3, model = "fdmo")
r3 <- fdmo$results
comparison <- data.frame(
farm = r3$id,
good_eff = round(r3$good_eff, 3),
paper_good = c(0, 0, 3, 0, 0.8),
bad_eff = round(r3$bad_eff, 3),
paper_bad = c(0, 0, 3.5, 0, 1.0),
maximal_y = round(r3$maximal_y, 3)
)
comparison
#> farm good_eff paper_good bad_eff paper_bad maximal_y
#> 1 A 0.000 0.0 0.000 0.0 10.000
#> 2 B 0.000 0.0 0.000 0.0 7.000
#> 3 C 3.000 3.0 3.500 3.5 10.000
#> 4 D 0.000 0.0 0.000 0.0 11.000
#> 5 E 0.789 0.8 0.921 1.0 10.789Farms A, B and D are on the frontier (gross inefficiency 0), and farm C expands its good output by 3 and cuts its bad output by 3.5, a gross inefficiency of 6.5: farms A to D reproduce Table 3 exactly. Farm E is the one divergence, shown side by side above. The paper prints \((\theta_y, \theta_b) = (0.8, 1.0)\); the package returns \((0.789, 0.921)\). The two cannot both be exact: whenever the uncontrolled emission equals controlled plus abatement, the materials-balance row forces \(\theta_b = v\,\theta_y\) for every farm, and \(7/6 \times 0.8 \approx 0.93\), not \(1.0\). The package’s pair satisfies the forced ratio to machine precision, so we read the printed row as rounded; farm E’s maximal good output still rounds to the paper’s 10.8.
Their Example 1 is a constant-returns technology with one input, one good output \(y\) and one bad output, written \(z\) in the paper (our \(b\)), solved analytically in the paper.
mrl <- pgt_tech(
x = matrix(c(1, 1, 1, 2, 2), ncol = 1, dimnames = list(NULL, "x")),
y = c(2, 3 / 2, 2 / 3, 3, 2),
b = c(4, 1, 2, 5, 3),
u = 1, polluting = "x",
id = paste0("DMU", 1:5)
)
bp <- pgt(mrl, model = "byprod", returns = "crs")
rb <- bp$results
data.frame(id = rb$id, output_eff = round(rb$output_eff, 4),
emission_eff = round(rb$efficiency, 4), fgl = round(rb$fgl, 4))
#> id output_eff emission_eff fgl
#> 1 DMU1 1.0000 0.2500 0.6250
#> 2 DMU2 0.7500 1.0000 0.8750
#> 3 DMU3 0.3333 0.5000 0.4167
#> 4 DMU4 0.7500 0.4000 0.5750
#> 5 DMU5 0.5000 0.6667 0.5833For DMU3 the output efficiency is \(1/3\), the emission efficiency \(1/2\) and the graph measure \(5/12\); for DMU2 they are \(3/4\), \(1\) and \(7/8\). These match the analytic scores in the paper.
This section is an internal-consistency check, not a replication of a printed table: the paper’s farm-level data are not published, so the four farms below are invented. What comes from the paper are the phosphorus contents (their footnote 16): feed 0.0124, piglet 0.0117 and meat 0.0117 kg P per kg. The check confirms that the implemented cost model decomposes environmental efficiency as \(EE = TE \times EAE\) with these coefficients; the model’s LPs are verified against hand-computed solutions in the test suite.
farms <- data.frame(
feed = c(200, 210, 190, 205),
piglet = c(20, 22, 18, 21),
meat = c(100, 100, 95, 102)
)
tech_c <- pgt_tech(
x = as.matrix(farms[, c("feed", "piglet")]),
y = farms$meat,
b = 0.0124 * farms$feed + 0.0117 * farms$piglet - 0.0117 * farms$meat,
u = c(feed = 0.0124, piglet = 0.0117),
v = 0.0117
)
mbc <- pgt(tech_c, model = "mb_cost", returns = "crs")
rc <- mbc$results
data.frame(id = rc$id, EE = round(rc$efficiency, 4),
TE = round(rc$te, 4), EAE = round(rc$eae, 4))
#> id EE TE EAE
#> 1 1 0.9955 1.0000 0.9955
#> 2 2 0.9442 0.9524 0.9914
#> 3 3 1.0000 1.0000 1.0000
#> 4 4 0.9885 0.9951 0.9934
all.equal(rc$efficiency, rc$te * rc$eae)
#> [1] TRUEThe identity \(EE = TE \times EAE\) holds to machine precision. (The identity holds by construction of the reported components, so this check guards the wiring, not the mathematics; the LP solutions themselves are checked by hand in the test suite.)