Help for package pgt

Type:

Package

Title:

Data Envelopment Analysis for Pollution-Generating Technologies

Version:

0.5.0

Description:

Nonparametric efficiency analysis for pollution-generating technologies under the materials-balance principle. Implements the weak-G-disposability model of Rodseth (2025) <doi:10.1007/s11123-025-00768-0> and its factorially determined multi-output representation, the by-production intersection technology of Murty, Russell and Levkoff (2012) <doi:10.1016/j.jeem.2012.02.005>, the materials-balance cost model of Coelli, Lauwers and Van Huylenbroeck (2007) <doi:10.1007/s11123-007-0052-8> and a weak-disposability reference model, with an enforced materials-balance identity, a pre-estimation feasibility audit, metafrontier decompositions, bad-output shadow prices, marginal abatement cost curves, a cross-axiom comparison harness, a global Malmquist-Luenberger productivity index and subsampling inference. Estimators are solved with 'lpSolveAPI'.

License:

GPL (≥ 3)

Encoding:

UTF-8

RoxygenNote:

7.3.3

Depends:

R (≥ 4.0.0)

Imports:

stats, graphics, utils, lpSolveAPI

Suggests:

testthat (≥ 3.0.0), knitr, rmarkdown, frontier

Config/testthat/edition:

VignetteBuilder:

knitr

LazyData:

true

URL:

https://github.com/iik1/pgt

BugReports:

https://github.com/iik1/pgt/issues

NeedsCompilation:

Packaged:

2026-07-18 10:22:12 UTC; s14454

Author:

Erik Enstad

[aut, cre]

Maintainer:

Erik Enstad <erik.enstad@nhh.no>

Repository:

CRAN

Date/Publication:

2026-07-28 15:50:02 UTC

pgt: Pollution-generating technologies under the materials balance

Description

Nonparametric efficiency analysis of pollution-generating technologies built on the materials-balance condition u'x - v y \ge b: the pollutant bound in a DMU's material inputs (u'x, with material flow coefficients u) is either retained in the good output (v y) or leaves as the bad output b. How bad outputs should enter the technology is contested, and the package implements the competing axiom systems (weak-G-disposability, by-production, materials-balance cost, weak disposability and the directional representation) behind one interface, so results can be compared across systems on identical data.

Details

The workflow: construct the technology with [pgt_tech()], audit the materials-balance accounts with [mb_check()], estimate with [pgt()], and post-process with [pgt_decompose()] (metafrontier decompositions), [shadow_prices()] and [mac_curve()] (output duals and marginal abatement costs) and [compare_models()] (cross-system agreement). [pgt_ml()] computes global Malmquist-Luenberger productivity change and [boot_pgt()] bootstrap inference.

Scope: each technology carries one good output (several pollutants and DMU-specific material flow coefficients are supported). The efficiency estimators treat the rows as an independent cross-section; panel structure enters only through [pgt_ml()]'s pooled global frontier, and [boot_pgt()] warns on panel technologies.

Author(s)

Maintainer: Erik Enstad erik.enstad@nhh.no (ORCID)

Subsampling inference for pollution-generating efficiency scores

Description

Constructs subsampling confidence intervals for the environmental efficiency scores of a pgt model and for group means, by re-solving each DMU against random subsamples of the reference set. Subsampling (m out of n, without replacement) is the resampling scheme with the firmest footing for the boundary-estimator, slow-convergence setting of nonparametric frontiers (Kneip, Simar and Wilson 2008; Simar and Wilson 2011).

Usage

boot_pgt(
  tech,
  model = c("wgd", "envelope", "byprod", "mb_cost", "wd"),
  B = 200,
  m = NULL,
  level = 0.95,
  kappa = NULL,
  returns = c("vrs", "crs"),
  peers = c("all", "group"),
  pollutant = 1L,
  seed = NULL
)

Arguments

tech

A [pgt_tech()] object.

model

One of the environmental-efficiency models ("wgd", "envelope", "byprod", "mb_cost", "wd"); the directional model "fdmo" is not supported.

B

Number of subsampling replicates.

m

Subsample size. Defaults to round(L^0.7), a point in the interior of the admissible range; Simar and Wilson (2011) show that sensitivity to m is the central practical issue, so check the default with [boot_pgt_sensitivity()].

level

Confidence level for the intervals.

kappa

Convergence-rate exponent used to rescale the subsample distribution. Defaults to the DEA rate 2/(d+1) under VRS and 2/d under CRS (Kneip, Simar and Wilson 2008) for the model's effective frontier dimension d: N+2 for "wgd" and "wd" (inputs, good and bad output), N+1 for "mb_cost" and "byprod" (whose programs live in the (x, y) and sub-technology spaces), and 2 for the input-free "envelope" model, whose frontier lives in the (y, b) plane. A heuristic; supply your own exponent to override.

returns, peers, pollutant

Passed to the underlying model, as in [pgt()].

seed

Optional integer seed for reproducibility. The caller's RNG state is restored on exit.

Details

The intervals follow the subsampling recipe of Politis and Romano (1994): the law of L^\kappa(\hat\theta - \theta) is approximated by the recentred, rate-scaled subsample distribution m^\kappa(\theta^*_m - \hat\theta), giving

[\hat\theta - (m/L)^\kappa q_{1-\alpha/2}(\theta^*_m - \hat\theta),\; \hat\theta - (m/L)^\kappa q_{\alpha/2}(\theta^*_m - \hat\theta)].

Each DMU is evaluated against the subsample frontier alone: the evaluated unit is not added to the reference set, so frontier units carry genuine resampling variation instead of a degenerate zero-width interval. Replicates in which a DMU's program is infeasible are dropped as NA and counted: per_dmu$n_ok reports the number of feasible replicates behind each interval, and a warning is issued when it falls below half of B. Because a subsample frontier lies weakly inside the full-sample frontier, the deviations are non-negative and the interval extends downward from the point estimate, matching the direction of the frontier bias; the point estimate itself typically lies at or above its own upper endpoint, since the interval targets the true score, which the estimate overestimates. The reported se is the subsample standard deviation rescaled by (m/L)^\kappa.

With peers = "group" the subsample is stratified: each group contributes in proportion to its size, with a floor of two units, so groups smaller than the floor are included whole and effectively not resampled.

Group-mean intervals are a descriptive companion, not a supported inference: the replicate statistic averages all group members against a common subsample frontier, so replicate variation reflects frontier movement only, and the per-DMU rate exponent is reused although means of frontier estimators converge at their own rates (see Kneip, Simar and Wilson 2015 for the aggregate theory). Read them as a stability band around the group mean, not as a confidence interval with nominal coverage.

The intervals remain heuristic. Subsampling theory for data envelopment analysis backs the radial and directional-distance technologies, not the materials-balance-constrained programs implemented here, so kappa borrows the DEA convergence rate as an approximation. Use the subsample-size sensitivity of [boot_pgt_sensitivity()] to check that the intervals are stable in m (its implied kappa_hat makes the borrowed rate checkable) before relying on them.

Value

An object of class "pgt_boot": a list with per_dmu (a data frame of point estimate, lower and upper bound, rescaled subsampling standard error, and n_ok, the number of feasible replicates, per DMU), group_means (the same for group means, when a group is present), and the call settings.

References

Kneip, A., Simar, L., & Wilson, P. W. (2008). Asymptotics and consistent bootstraps for DEA estimators in nonparametric frontier models. Econometric Theory, 24(6), 1663–1697. doi:10.1017/S0266466608080651

Kneip, A., Simar, L., & Wilson, P. W. (2015). When bias kills the variance: Central limit theorems for DEA and FDH efficiency scores. Econometric Theory, 31(2), 394–422. doi:10.1017/S0266466614000413

Politis, D. N., & Romano, J. P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. The Annals of Statistics, 22(4), 2031–2050. doi:10.1214/aos/1176325770

Politis, D. N., Romano, J. P., & Wolf, M. (1999). Subsampling. Springer, New York. doi:10.1007/978-1-4612-1554-7

Simar, L., & Wilson, P. W. (2011). Inference by the m out of n bootstrap in nonparametric frontier models. Journal of Productivity Analysis, 36(1), 33–53. doi:10.1007/s11123-010-0200-4

Examples

data(steeldemo)
steel60 <- steeldemo[1:60, ]
tech <- pgt_tech(
  x = steel60[, c("coal_coke", "other_fuel", "raw_material", "flux")],
  y = steel60$production, b = steel60$emissions, v = 0.01467,
  group = steel60$route, id = steel60$plant
)

bt <- boot_pgt(tech, model = "wgd", B = 40, seed = 1)
head(bt$per_dmu)

Subsample-size sensitivity of subsampling intervals

Description

Re-runs [boot_pgt()] over a grid of subsample sizes m and reports how the median interval width responds. Stable widths across the grid support the choice of m; a strong trend warns that the intervals are driven by the resampling size rather than the data.

Usage

boot_pgt_sensitivity(
  tech,
  model = "wgd",
  B = 100,
  m_grid = NULL,
  level = 0.95,
  kappa = NULL,
  returns = "vrs",
  peers = "all",
  pollutant = 1L,
  seed = NULL
)

Arguments

tech

A [pgt_tech()] object.

model, B, level, kappa, returns, peers, pollutant, seed

As in [boot_pgt()].

m_grid

Integer vector of subsample sizes. Defaults to five sizes spanning L^{0.5} to L^{0.9}.

Details

When at least three grid points have positive widths, the attribute "kappa_hat" reports the convergence-rate exponent implied by the log-spread regression of Politis, Romano and Wolf (1999): the rescaled width scales as m^{\kappa - \kappa_0} for true rate \kappa_0 and working rate \kappa, so \hat\kappa_0 = \kappa - slope of \log width on \log m. Agreement between kappa_hat and the working kappa (attribute "kappa_used") supports the borrowed DEA rate; disagreement suggests supplying kappa_hat to [boot_pgt()]. The same seed is reset before every grid point, so replicate draws are comparable across m.

Value

A data frame with one row per m: the size, the median per-DMU interval width, the median over strictly positive widths, and the mean rescaled standard error, with attributes "kappa_used" and (when estimable) "kappa_hat".

Examples

data(steeldemo)
steel60 <- steeldemo[1:60, ]
tech <- pgt_tech(
  x = steel60[, c("coal_coke", "other_fuel", "raw_material", "flux")],
  y = steel60$production, b = steel60$emissions, v = 0.01467,
  id = steel60$plant
)

boot_pgt_sensitivity(tech, model = "wgd", B = 25,
                     m_grid = c(15, 25, 40), seed = 1)

Compare competing axiom systems on identical data

Description

Fits several pollution-generating technology models to the same [pgt_tech()] object and reports how their environmental efficiency scores and rankings agree. The materials-balance debate turns on which disposability axiom is appropriate for bad outputs (Forsund 2021 and the accompanying Journal of Productivity Analysis symposium); this function puts the competing systems side by side so the sensitivity of conclusions to that choice is visible.

Usage

compare_models(
  tech,
  models = c("wgd", "byprod", "mb_cost", "wd"),
  returns = c("vrs", "crs"),
  peers = c("all", "group"),
  pollutant = 1L
)

Arguments

tech

A [pgt_tech()] object.

models

Character vector of models to compare. Defaults to c("wgd", "byprod", "mb_cost", "wd"): the weak-G-disposability (materials balance), by-production, materials-balance cost and weak-disposability systems.

returns

Returns to scale passed to every model: "vrs" (default) or "crs".

peers

Reference set passed to every model: "all" (default) or "group".

pollutant

For a multi-pollutant technology, the pollutant to compare on. Defaults to the first.

Details

Each model reports its headline environmental-efficiency score, normalised so that 1 is efficient: b^*/b for "wgd", "envelope", "byprod" and "wd", and the material-inflow ratio EE = u'x^*/u'x for "mb_cost". The directional model "fdmo" stays excluded, since its score is a gross inefficiency on another scale. Because the scores measure different quantities, only the rank-based statistics (the Spearman matrix and the bottom-quartile overlap) are strictly comparable across models; the median column is a per-model summary. Rank agreement is measured by Spearman correlation over the DMUs solved by both members of each pair. Note that "wgd" scores can exceed 1 for DMUs violating another pollutant's materials-balance identity (see [pgt()]); such DMUs enter the comparison unflagged.

Value

An object of class "pgt_compare": a list with

scores: Data frame of environmental efficiency, one column per model, one row per DMU (with id and group).
spearman: Matrix of pairwise Spearman rank correlations.
agreement: Data frame with, per model, the number of DMUs solved, the median score, and bottom_q_overlap: the share of the DMUs this model places in its own bottom efficiency quartile that the reference model (the first entry of models) also places in its bottom quartile (top-of-agenda overlap). The bottom quartile includes all DMUs tied at the 25 per cent cutoff, so it can exceed a quarter of the sample and its size can differ across models.
models, returns, peers: Call settings.

References

Forsund, 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

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

Examples

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
)
cmp <- compare_models(tech, models = c("wgd", "byprod", "mb_cost"))
cmp

Marginal abatement cost curve

Description

Builds a marginal abatement cost (MAC) curve from a [pgt()] fit. Along the estimated frontier, reducing the bad output by one unit requires forgoing 1 / \eta_l units of the good output, where \eta_l = \partial b^* / \partial y is the output-constraint dual. The DMU's marginal abatement cost is therefore p / \eta_l for an output price p; with no price supplied the curve is expressed in output units per unit of bad output (1 / \eta_l). Each DMU's abatement potential is its distance to the frontier, b_l - b^*_l. DMUs are ordered by increasing marginal cost and the potential is accumulated.

Usage

mac_curve(fit, price = NULL)

Arguments

fit

A [pgt()] fit with model = "wgd" or "envelope", the models that report output duals.

price

Optional scalar price of the good output. If supplied, mac is in money per unit of bad output.

Details

The curve therefore combines two margins. It orders DMUs by the shadow price at their frontier projection and plots against it the abatement available from eliminating inefficiency (the gap b_l - b^*_l), which the model itself prices at zero output loss; the mac value is the marginal cost of abatement beyond the frontier point. The area under the curve is consequently not a total-cost estimate.

DMUs with \eta_l = 0 are excluded: a zero output dual arises when the output constraint is slack, i.e. the DMU projects onto the flat segment of the (y, b) frontier, where abatement via output contraction has locally infinite marginal cost. Only exact-zero duals are excluded. Unsolved LPs are excluded as well. The number of excluded DMUs and the summed abatement potential of the solved-but-excluded ones (which is NOT part of the curve) are attached as attributes "n_excluded" and "excluded_abatement" and reported by print.

A DMU that projects onto a vertex of the piecewise-linear frontier has a range of optimal duals, and the solver reports the dual of whichever optimal basis it terminates in, so the MAC of such a DMU is one point of an interval and can differ across solver versions or platforms. Frontier-interior projections have unique duals.

Value

A data frame of class "pgt_mac", ordered by mac: id, group (if present), mac, abatement (b - b^*) and cum_abatement.

References

Fare, R., Grosskopf, S., Lovell, C. A. K., & Yaisawarng, S. (1993). Derivation of shadow prices for undesirable outputs: A distance function approach. The Review of Economics and Statistics, 75(2), 374–380. doi:10.2307/2109448

Rodseth, 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

Examples

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
)
fit <- pgt(tech, model = "wgd")
mac <- mac_curve(fit, price = 550)
plot(mac)

Audit the materials-balance identity

Description

Pre-estimation feasibility audit of the materials-balance condition u'x_l - v y_l \ge b_l for every DMU (and every pollutant). DMUs that violate the identity carry inconsistent material accounts (more pollutant leaves than enters). A violation makes self-reference infeasible in the weak-G-disposability program, so that DMU's LP solves only if some peer mix meets every constraint within its materials-balance cap; conversely, infeasibility is confined to DMUs with an exact violation, gap < 0, in at least one pollutant account (pollutant p's LP can be infeasible because of a violation in a different pollutant q; DMUs satisfying every account always solve via self-reference). Audit before estimation.

Usage

mb_check(tech, tol = 1e-08)

Arguments

tech

A [pgt_tech()] object.

tol

Relative tolerance: a DMU-pollutant account is flagged (violated) when its closure gap is below -tol times its pollutant potential u'x_l. The flag is a data-quality screen; LP infeasibility tracks the exact sign of the gap, so the flagged set is a subset of the accounts with gap < 0 (counted separately in attribute "n_exact").

Value

A data frame of class "pgt_mb" with one row per DMU (per pollutant when several are present): id, group (if present), pollutant (when several), potential (u'x_l), retained (v y_l), b, gap (u'x_l - v y_l - b_l), rel_gap (gap / potential) and violated. When the technology records an abatement output, also a and closure (gap - a_l): the equality residual u'x_l - v y_l - b_l - a_l, zero when the account closes exactly, which model = "fdmo" requires. The number of violations is attached as attribute "n_violations".

Examples

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
)
mb <- mb_check(tech)
mb

Estimate a pollution-generating technology model

Description

Fits a nonparametric materials-balance efficiency model by solving one linear program per DMU. The minimising models return the least bad output b^*_l attainable at the DMU's activity level, with environmental efficiency b^*_l / b_l \in (0, 1]; the directional model returns the joint good-output expansion and bad-output contraction.

Usage

pgt(
  tech,
  model = c("wgd", "wgd_rodseth", "envelope", "fdmo", "ddf", "byprod", "mb_cost", "wd"),
  returns = c("vrs", "crs"),
  peers = c("all", "group"),
  pollutant = 1L
)

Arguments

tech

A [pgt_tech()] object.

model

Character: "wgd" (alias "wgd_rodseth"), "envelope", "fdmo" (alias "ddf"), "mb_cost", "byprod" or "wd". See Details.

returns

Returns to scale: "vrs" (default) or "crs".

peers

Reference set: "all" pools every DMU; "group" restricts each DMU's peers to its own technology group (requires group in [pgt_tech()]).

pollutant

For a multi-pollutant technology, the pollutant to estimate: a column name or index of b. Defaults to the first pollutant. This selects which bad output the objective acts on. For model = "wgd" the materials-balance caps of every pollutant constrain the projection; the other models use only the selected pollutant's data.

Details

The models fall into two axiom families. The materials-balance models ("wgd", "envelope", "fdmo", "mb_cost") enforce the identity u'x_l - v y_l \ge b_l; the reference models ("byprod", "wd") implement competing systems for cross-comparison (see [compare_models()]).

"wgd": The weak-G-disposability model of Rodseth (2025), Eq. 6 in reduced form: minimise the bad output subject to output and input feasibility, the peer emission envelope, and the evaluated DMU's own materials-balance cap b \le u'x_l - v y_l. With several pollutants the caps of the remaining pollutants constrain the peer mix as well, so the projection respects every pollutant's materials balance. DMUs whose data violate a materials-balance identity lose the self-reference guarantee: their LP may be infeasible (returning NA), and every infeasible LP belongs to such a DMU. A DMU violating another pollutant's identity can also score above 1, since its MB-consistent projection may emit more of the selected pollutant than the DMU reports. Audit with [mb_check()] first. model = "wgd_rodseth" is an alias.
"envelope": Rodseth's Eq. 6 with inputs free: the program reduces to the convex lower envelope of the (y, b) scatter (the weak-G slack equality is absorbed by the input reallocation). Self-reference is always feasible, so all scores lie in (0, 1] with no feasibility screen.
"fdmo": The factorially determined multi-output (directional) representation of Rodseth (2025), Eq. 13 with the abatement output fixed at each DMU's own level. It jointly maximises the expansion of the good output (\theta_y) and the contraction of the bad output (\theta_b) along the materials-balance frontier; gross inefficiency is \theta_y + \theta_b, with 0 for a frontier DMU. The gross score adds good-output units to bad-output units (direction (1, 1)), so it is unit-dependent and comparable only within one unit system, as in Rodseth's Table 3. Requires an abatement output a in [pgt_tech()]. The model imposes the materials-balance identity as an exact equality u'x_l - v y_l = b_l + a_l: accounts that do not close exactly either make the LP infeasible or shift the closure gap into the scores, and pgt() warns when it detects open accounts. model = "ddf" is an alias.
"mb_cost": The materials-balance cost model of Coelli, Lauwers and Van Huylenbroeck (2007): environmental efficiency is the ratio of minimal to observed aggregate material inflow u'x, decomposed as EE = TE \times EAE into technical and environmental-allocative efficiency (returned as te and eae).
"byprod": The by-production intersection technology of Murty, Russell and Levkoff (2012): the good output is produced by one sub-technology and the bad output generated by another over the emission-causing inputs. The implemented model is the original Murty-Russell-Levkoff intersection technology without integrated abatement; Hampf (2014) develops the abatement-integrated extension. Returns emission efficiency (b^*/b, the headline), output efficiency (output_eff) and fgl, the arithmetic mean of the two sub-efficiencies, in the spirit of the Fare-Grosskopf-Lovell graph measure. Set the emission-causing inputs with polluting in [pgt_tech()].
"wd": The weak-disposability model (Kuosmanen 2005 correct VRS formulation; see the exchange settled in Kuosmanen and Podinovski 2009), a reference axiom system that imposes no materials balance. Returns emission efficiency b^*/b.

Value

An object of class "pgt": a list with

results: Data frame with one row per DMU. For "wgd" and "envelope": id, group (if present), y, b, b_star, efficiency (b^*/b), dual_output (shadow value of the output constraint, \partial b^*/\partial y \ge 0), mb_headroom (weak-G-disposability only: the slack u'x_l - v y_l - b^*_l; NA for the envelope model) and status (0 = solved). For "fdmo": id, group (if present), y, b, gross (\theta_y + \theta_b), good_eff (\theta_y), bad_eff (\theta_b), maximal_y (y_l + \theta_y) and status.
weights: List of named vectors of non-zero peer weights per DMU, keyed by make.unique(id).
model, returns, peers, pollutant: The call settings.

References

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

Fare, R., Grosskopf, S., & Lovell, C. A. K. (1985). The Measurement of Efficiency of Production. Kluwer-Nijhoff, Boston. doi:10.1007/978-94-015-7721-2

Hampf, B. (2014). Separating environmental efficiency into production and abatement efficiency: A nonparametric model with application to US power plants. Journal of Productivity Analysis, 41(3), 457–473. doi:10.1007/s11123-013-0357-8

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

Kuosmanen, T., & Podinovski, V. V. (2009). Weak disposability in nonparametric production analysis: Reply to Fare and Grosskopf. American Journal of Agricultural Economics, 91(2), 539–545. doi:10.1111/j.1467-8276.2008.01238.x

Examples

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
)
fit <- pgt(tech, model = "wgd")
summary(fit)

Decompose environmental efficiency across technology groups

Description

Metafrontier decompositions of the environmental efficiency score b^*/b into within-group and technology-gap components, built on Rodseth's (2025) weak-G-disposability technology and the DEA metafrontier and technology-gap ratio of O'Donnell, Rao and Battese (2008); the metafrontier idea originates with Battese, Rao and O'Donnell (2004).

Usage

pgt_decompose(
  tech,
  type = c("envelope", "rodseth"),
  returns = c("vrs", "crs"),
  pollutant = 1L
)

Arguments

tech

A [pgt_tech()] object with a non-NULL group.

type

"envelope" (default) or "rodseth". See Details.

returns

Returns to scale: "vrs" (default) or "crs".

pollutant

For a multi-pollutant technology, the pollutant to decompose: a column name or index of b. Defaults to the first.

Details

"envelope"

The (y, b)-envelope decomposition built on Rodseth's (2025) Eq. 6 with inputs free. Stages differ by peer set only, so the weak-G-disposability slack equality is respected throughout:

Total = \frac{b^*_{all}}{b} = WR \times TGR, \quad WR = \frac{b^*_{group}}{b}, \quad TGR = \frac{b^*_{all}}{b^*_{group}}.

WR is within-group reallocation efficiency; TGR is the technology-gap ratio (O'Donnell, Rao and Battese 2008). Self-reference is always feasible, so every component lies in (0, 1] with no feasibility screen.

"rodseth"

A three-stage decomposition of the full weak-G-disposability model (Rodseth 2025, Eq. 11, collapsed to three components when there are no dedicated abatement inputs):

EnvEff = \frac{b^*_3}{b} = \underbrace{\frac{b^*_1}{b}}_{TE} \times \underbrace{\frac{b^*_2}{b^*_1}}_{Technology} \times \underbrace{\frac{b^*_3}{b^*_2}}_{AE},

where stage 1 solves the WGD program against own-group peers (technical efficiency), stage 2 against all peers (technology gap), and stage 3 additionally drops the input constraints (input-mix / allocative component). Each stage is a strict relaxation of the previous one, so every component lies in (0, 1] where feasible for DMUs satisfying every pollutant's materials-balance identity; as in [pgt()], a DMU violating another pollutant's identity can have te and total above 1, since its MB-consistent projection may emit more of the selected pollutant than the DMU reports. When an early stage is infeasible (materials-balance violators, see [mb_check()]) its component is NA while later-stage ratios and total can remain defined, matching the reference implementation; the multiplicative identity holds for rows with all components present.

Value

An object of class "pgt_decomp": a list with results (one row per DMU with the stage minima and the multiplicative components), type and returns. The component columns are WR, TGR, total for type = "envelope" and te, technology, ae, total for type = "rodseth".

References

Battese, G. E., Rao, D. S. P., & O'Donnell, C. J. (2004). A metafrontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies. Journal of Productivity Analysis, 21(1), 91–103. doi:10.1023/B:PROD.0000012454.06094.29

O'Donnell, C. J., Rao, D. S. P., & Battese, G. E. (2008). Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Economics, 34(2), 231–255. doi:10.1007/s00181-007-0119-4

Examples

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
)
dec <- pgt_decompose(tech, type = "envelope")
summary(dec)

Global Malmquist-Luenberger productivity index

Description

Computes the global Malmquist-Luenberger (GML) productivity index of Oh (2010) for a panel of pollution-generating technologies, decomposed into efficiency change and best-practice (frontier) change. The index is built on the directional distance function of the pollution- generating technology, which expands the good output and contracts the bad output jointly, so productivity change accounts for emissions rather than ignoring them.

Usage

pgt_ml(
  tech,
  technology = c("wgd", "envelope", "wd"),
  returns = c("vrs", "crs"),
  pollutant = 1L
)

Arguments

tech

A [pgt_tech()] object with a non-NULL period and id identifying the same DMU across periods.

technology

Reference technology: "wgd" keeps the input constraints and the lower emission envelope (default); "envelope" frees the inputs, so the frontier becomes the (y, b) envelope; "wd" imposes weak disposability of the bad output, the technology of Oh (2010). See Details.

returns

Returns to scale: "vrs" (default) or "crs".

pollutant

For a multi-pollutant technology, the pollutant to index. Defaults to the first.

Details

The directional distance to a reference technology R is

D_R(x, y, b) = \max\{\beta : (y + \beta g_y, b - \beta g_b) \textrm{ feasible in } R \textrm{ given } x\},

with direction g = (y, b) (each observation is scaled by its own level). The global reference pools every period into one technology (Oh 2010), which removes the cross-period infeasibility of the adjacent-period Malmquist-Luenberger index (Aparicio, Pastor and Zofio 2013) and makes the index circular. For a DMU observed in periods t and t+1,

GML = \frac{1 + D_G(t)}{1 + D_G(t+1)} = \underbrace{\frac{1 + D_t(t)}{1 + D_{t+1}(t+1)}}_{EC} \times \underbrace{GML / EC}_{BPC},

where D_G is the distance to the global frontier and D_t the distance to period t's own frontier. GML > 1 is productivity growth; EC is catch-up to the frontier and BPC is frontier movement.

Three reference technologies are available. "wd" imposes weak disposability of the bad output (the Kuosmanen 2005 VRS form, with the intensity weights split into an active and an abatement part): this is the technology under which Chung, Fare and Grosskopf (1997) and Oh (2010) define the (global) Malmquist-Luenberger index, so technology = "wd" is the faithful Oh (2010) comparator. "wgd" (the default) and "envelope" instead treat the bad output as reducible down to the peer emission envelope with no proportional output sacrifice, the same lower-envelope treatment as the corresponding [pgt()] programs, with ("wgd") and without ("envelope") the input constraints. Under constant returns the free-disposal technologies contain the weak-disposability set, so their distances are weakly larger than under "wd"; under variable returns the sets are not nested in general, because the Kuosmanen form lets the active intensity weights sum below one.

This estimator is experimental: the global Malmquist-Luenberger index under a materials-balance technology is not yet settled in the literature. Under "wgd" and "envelope" the index does not re-impose the per-DMU materials-balance cap on the projected point, and the bad output is freely disposable toward the envelope rather than weakly disposable, so these variants are exploratory companions to the "wd" index, not implementations of Oh (2010).

Value

An object of class "pgt_ml": a list with results, a data frame with one row per DMU per consecutive period transition (id, from, to, gml, ec, bpc), and the call settings. Transitions are formed only for DMUs present in both adjacent periods.

References

Chambers, R. G., Chung, Y., & Fare, R. (1996). Benefit and distance functions. Journal of Economic Theory, 70(2), 407–419. doi:10.1006/jeth.1996.0096

Chung, Y. H., Fare, R., & Grosskopf, S. (1997). Productivity and undesirable outputs: A directional distance function approach. Journal of Environmental Management, 51(3), 229–240. doi:10.1006/jema.1997.0146

Oh, D.-h. (2010). A global Malmquist-Luenberger productivity index. Journal of Productivity Analysis, 34(3), 183–197. doi:10.1007/s11123-010-0178-y

Aparicio, J., Pastor, J. T., & Zofio, J. L. (2013). On the inconsistency of the Malmquist-Luenberger index. European Journal of Operational Research, 229(3), 738–742. doi:10.1016/j.ejor.2013.03.031

Examples

# Two-period panel: period 2's frontier emits less for the same output.
set.seed(1)
n <- 12
d1 <- data.frame(id = 1:n, y = runif(n, 5, 10), x = runif(n, 8, 12),
                 b = runif(n, 2, 6))
d2 <- data.frame(id = 1:n, y = d1$y * 1.05, x = d1$x, b = d1$b * 0.85)
d <- rbind(cbind(d1, period = 1), cbind(d2, period = 2))
tech <- pgt_tech(x = d[, "x", drop = FALSE], y = d$y, b = d$b,
                 period = d$period, id = d$id)
ml <- pgt_ml(tech)
summary(ml)

Construct a pollution-generating technology

Description

Bundles the data of a pollution-generating technology, a single good output, one or more bad outputs, and material inputs, together with the material flow coefficients that tie them to the materials-balance identity. The returned object is the input to [pgt()], [pgt_decompose()] and [mb_check()].

Usage

pgt_tech(
  x,
  y,
  b,
  u = NULL,
  v = 0,
  a = NULL,
  x_abate = NULL,
  polluting = NULL,
  group = NULL,
  period = NULL,
  id = NULL
)

Arguments

x

Numeric matrix or data frame of material inputs, one row per decision-making unit (DMU), one column per input.

y

Numeric vector of the good output, strictly positive.

b

Bad outputs (controlled emissions), strictly positive: a numeric vector for a single pollutant, or an L \times P matrix or data frame with one column per pollutant.

u

Material flow coefficients of the inputs. A length-N vector (one coefficient per input column, shared by all DMUs), an L \times N matrix (DMU-specific coefficients), or, with several pollutants, a named list with one such element per pollutant. Defaults to 1 for every input and pollutant.

v

Material flow coefficient of the good output (pollutant embodied in the product). A scalar, a length-L vector (DMU-specific), or, with several pollutants, a length-P vector, L \times P matrix, or list. Defaults to 0.

a

Optional abatement output (pollutant removed by end-of-pipe control), non-negative: a numeric vector for a single pollutant or an L \times P matrix. Required by the models that treat pollution control explicitly; see [pgt()].

x_abate

Optional marker of the pollution-control input columns of x (Rodseth 2025, Eq. 7): a character vector of column names or an integer vector of column positions. The remaining columns are production inputs. Currently recorded and reported only: the estimators do not yet treat pollution-control inputs separately, and [pgt()] warns when the marker is set.

polluting

Optional marker of the emission-generating input columns of x (the residual-generating sub-technology of the by-production model; Murty, Russell and Levkoff 2012): a character vector of column names or an integer vector of column positions. Defaults, when the by-production model is fitted, to the inputs with a positive material flow coefficient.

group

Optional factor (or vector coercible to factor) assigning each DMU to a technology group, for example a production route. Required by [pgt_decompose()] and by group-referenced estimation.

period

Optional factor (or vector coercible to factor) giving the time period of each row, for panel data. Required by [pgt_ml()]. The pair (id, period) identifies an observation.

id

Optional vector of DMU identifiers. Defaults to the row names of x, or a running index.

Details

The materials-balance principle requires that the pollutant content of the inputs is either embodied in the good output, removed by end-of-pipe abatement, or emitted:

u'x_l - v y_l \ge b_l,

where u holds the material flow coefficients of the inputs (for example tonnes of CO2 potential per unit of input) and v the coefficient of the good output (pollutant retained in the product). Data already expressed in pollutant-potential units use the default u = 1 for every input. When the abatement output a is observed, the identity closes as an equality, u'x_l - v y_l = b_l + a_l. Use [mb_check()] to audit the identity before estimation.

Material flow coefficients may vary across DMUs (heterogeneous input or output quality; Eder 2022, Rodseth 2025). Supply u as an L \times N matrix and v as a length-L vector to attach DMU-specific coefficients. With homogeneous coefficients a length-N vector u and scalar v suffice, and all estimators reduce to the homogeneous-quality case.

With several pollutants, supply b as an L \times P matrix (one named column per pollutant), u as a named list with one element per pollutant (each a length-N vector or L \times N matrix), and v as a length-P vector, L \times P matrix, or list. Each pollutant carries its own materials-balance identity.

Value

An object of class "pgt_tech": a list with elements x (L \times N matrix), y (length L), b (L \times P matrix), u (L \times N \times P array), v (L \times P matrix), a (L \times P matrix or NULL), x_abate (integer vector of pollution-control input columns, possibly empty), group, id, L, N, P and pollutants (character vector of pollutant names). Single-pollutant input is stored in the same canonical shapes with P = 1.

References

Eder, A. (2022). Environmental efficiency measurement when producers control pollutants under heterogeneous conditions: a generalization of the materials balance approach. Journal of Productivity Analysis, 57(2), 157–176. doi:10.1007/s11123-021-00623-y

Examples

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
)
tech

# Explicit abatement: Rodseth (2025) pig-finishing example, with the
# uncontrolled-emission aggregate as the material carrier
data(pigfarms)
tech2 <- 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
)

Pig-finishing farms from Rodseth (2025), Table 1

Description

The synthetic manure-transport example printed as Table 1 of Rodseth (2025): five pig-finishing farms with piglets and feed as material inputs, labour and capital as non-material inputs, saleable meat as the good output, and nitrogen emissions as the bad output, extended with end-of-pipe abatement. The example descends from Coelli, Lauwers and Van Huylenbroeck (2007) via Rodseth (2016). The columns satisfy uncontrolled - controlled = abatement exactly.

Usage

pigfarms

Format

A data frame with 5 rows and 9 columns:

farm: Farm identifier, "A" to "E".
feed: Feed input.
piglet: Piglet input.
labor: Labour input.
capital: Capital input.
meat: Saleable meat (good output).
controlled: Controlled nitrogen emissions (bad output).
uncontrolled: Uncontrolled (ex ante) nitrogen emissions.
abatement: End-of-pipe abatement, uncontrolled - controlled.

Details

The paper does not print the nitrogen coefficients of feed and piglets, so the technology is constructed in pollutant-potential units: the uncontrolled-emission aggregate carries the material with u = 1 on that column and v = 0, which reproduces the paper's materials-balance cap exactly. With this mapping, pgt(tech, model = "wgd") reproduces the minimal controlled emissions of the paper's Table 2 (16, 16, 16, 20, 16), and [mb_check()]'s closure gap recovers the abatement column.

Source

References

Rodseth, K. L. (2016). Environmental efficiency measurement and the materials balance condition reconsidered. European Journal of Operational Research, 250(1), 342–346. doi:10.1016/j.ejor.2015.10.061

Examples

data(pigfarms)
tech <- pgt_tech(
  x = pigfarms[, c("feed", "piglet", "labor", "capital", "uncontrolled")],
  y = pigfarms$meat,
  b = pigfarms$controlled,
  u = c(0, 0, 0, 0, 1),
  v = 0,
  id = pigfarms$farm
)
# closure gap recovers the paper's abatement column
mb_check(tech)$gap
# minimal controlled emissions of Rodseth (2025), Table 2
pgt(tech, model = "wgd")$results$b_star

Extract shadow values from a pgt fit

Description

Returns the informative constraint diagnostics of the per-DMU linear programs. dual_output is the shadow value of the output constraint, \partial b^* / \partial y \ge 0: the marginal bad-output content of the good output along the frontier (for CO2, the marginal emission intensity in tonnes of CO2 per tonne of output). mb_headroom (weak-G-disposability model only) is the slack u'x_l - v y_l - b^*_l between the projection and the DMU's materials-balance ceiling.

Usage

shadow_prices(fit)

Arguments

fit

A [pgt()] fit with model = "wgd" or "envelope", the models that report output duals.

Details

The dual of the peer emission envelope \sum_l \lambda_l b_l \le b is not reported: in this reduced form it equals -1 whenever the LP solves (dual feasibility on the emission variable, with the materials-balance row slack at any optimum), so it carries no cross-DMU information. In the boundary case b^* = u'x_l - v y_l exactly, the split of this dual between the envelope and cap rows is basis-dependent. A materials-balance cap that would bind manifests as an infeasible LP, not as a capped projection.

Values are reported in the units of the linear program (quantities, not money). To monetise the output dual, combine it with an output price via [mac_curve()].

Value

A data frame with one row per DMU: id, group (if present), b, b_star, dual_output and mb_headroom (NA for the envelope model).

References

Examples

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
)
fit <- pgt(tech, model = "wgd")
head(shadow_prices(fit))

Synthetic steel plant panel

Description

A simulated panel of steel plants loosely calibrated to the structure of global plant-level data: two production routes with different carbon intensities, four material inputs expressed in CO2-potential units (so the material flow coefficients are u = 1), crude steel output, and Scope 1 CO2 emissions satisfying the materials-balance identity u'x - v y \ge b with v = 0.01467 (carbon retained in the product). The synthetic generator (see data-raw/steeldemo.R) assumes roughly 0.4 per cent retained carbon by mass, converted to CO2 units: 0.004 \times 44/12 = 0.01467 tonnes of CO2 per tonne of steel; emissions are drawn as the CO2 potential minus this retained content, minus a small non-emitted remainder, so the identity holds in every row. The data are synthetic; they mimic magnitudes, not any real plant.

Usage

steeldemo

Format

A data frame with 180 rows (60 plants over 3 years) and 9 columns:

plant: Plant identifier.
year: Observation year.
route: Production route: "Integrated" or "Minimill".
production: Crude steel output (tonnes).
coal_coke: Coal and coke input (tonnes CO2 potential).
other_fuel: Other fuel input (tonnes CO2 potential).
raw_material: Raw material input (tonnes CO2 potential).
flux: Flux and alloy input (tonnes CO2 potential).
emissions: Scope 1 CO2 emissions (tonnes).

Source

Simulated; see data-raw/steeldemo.R in the package sources.

Package {pgt}

pgt: Pollution-generating technologies under the materials balance

Description

Details

Author(s)

See Also

Subsampling inference for pollution-generating efficiency scores

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Subsample-size sensitivity of subsampling intervals

Description

Usage

Arguments

Details

Value

See Also

Examples

Compare competing axiom systems on identical data

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Marginal abatement cost curve

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Audit the materials-balance identity

Description

Usage

Arguments

Value

See Also

Examples

Estimate a pollution-generating technology model

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Decompose environmental efficiency across technology groups

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Global Malmquist-Luenberger productivity index

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Construct a pollution-generating technology

Description

Usage

Arguments

Details