pgt: Data Envelopment Analysis for Pollution-Generating Technologies

pgt implements nonparametric efficiency analysis for pollution-generating technologies under the materials-balance principle. It packages the competing axiom systems for modelling bad outputs behind one interface, with an enforced materials-balance identity, a pre-estimation feasibility audit, metafrontier decompositions, bad-output shadow prices, marginal abatement cost curves, a productivity index and subsampling inference.

The materials-balance principle goes back to Ayres and Kneese (1969); Lauwers (2009) makes the case for building it into frontier models, and Dakpo, Jeanneaux and Latruffe (2016) survey the modelling landscape. Existing R packages handle undesirable outputs by data translation (deaR's Seiford-Zhu undesirable-output models) or by weak disposability and directional distances (Benchmarking::dea.direct, nonparaeff's directional-distance routines), but, to our knowledge, no maintained package in R or any other major statistical ecosystem ships materials-balance-constrained DEA; applied studies implement these estimators in ad-hoc optimisation code. pgt packages them behind one coherent API.

Scope

Each technology has one good output; several pollutants (with their own materials-balance accounts) 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 when a panel technology is subsampled.

Installation

# development version
remotes::install_github("iik1/pgt")
# the peer-reviewed snapshot
remotes::install_github("iik1/pgt@v0.4.1")

Cite the package with citation("pgt").

Quick start

library(pgt)
data(steeldemo)   # synthetic steel-plant panel shipped with the package

# 1. Build the technology: inputs in CO2-potential units (u = 1),
#    v = carbon retained in the product.
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
)

# 2. Audit the materials-balance identity before estimating.
mb_check(tech)

# 3. Fit the weak-G-disposability model (Rodseth 2025, Eq. 6).
fit <- pgt(tech, model = "wgd")
summary(fit)

# 4. Decompose environmental efficiency across production routes.
summary(pgt_decompose(tech, type = "envelope"))

# 5. Shadow prices and the marginal abatement cost curve.
head(shadow_prices(fit))
plot(mac_curve(fit, price = 550))

# 6. Compare competing axiom systems on the same data.
compare_models(tech, models = c("wgd", "byprod", "mb_cost", "wd"))

Models

Function	What it does
pgt_tech()	Technology constructor: inputs, good/bad outputs, material flow coefficients u, v, abatement a, technology groups, panel period
mb_check()	Audit of u'x - v y >= b per DMU and pollutant
pgt(model = "wgd")	Rodseth (2025) weak-G-disposability LP
pgt(model = "envelope")	Eq. 6 with inputs free: convex lower (y, b) envelope
pgt(model = "fdmo")	Rodseth (2025) directional representation, Eq. 13 (alias "ddf")
pgt(model = "mb_cost")	Coelli et al. (2007) materials-balance cost model, EE = TE x EAE
pgt(model = "byprod")	Murty-Russell-Levkoff (2012) by-production intersection technology
pgt(model = "wd")	Kuosmanen (2005) weak-disposability reference model
pgt_decompose()	Metafrontier decompositions (envelope WR x TGR; staged Rodseth)
compare_models()	Competing axiom systems on identical data, with rank agreement
pgt_ml()	Global Malmquist-Luenberger productivity index (Oh 2010; experimental)
boot_pgt()	Subsampling inference for scores and group means (heuristic intervals)
shadow_prices(), mac_curve()	Constraint duals and marginal abatement cost curves

DMU-specific material flow coefficients and multiple pollutants are supported: pass u as a matrix or list and b as a matrix.

Validation

Validation runs as unit tests on every check, at three levels of strength.

• Published-table replication. pgt(model = "wgd") reproduces the Rødseth (2025) Table 2 minimal controlled emissions (16, 16, 16, 20, 16), pgt(model = "fdmo") the Table 3 directional scores for farms A to D exactly (farm E is discussed in the replication vignette), and pgt(model = "byprod") the analytic efficiency scores of Murty, Russell and Levkoff's (2012) Example 1. The package ships the pig-finishing example as data(pigfarms).
• Analytic hand-computed checks. The mb_cost, wd and envelope models are verified against small problems solved by hand, and the wgd kernel against an independent reference implementation in the test suite. The mb_cost check uses the phosphorus material-flow coefficients of Coelli, Lauwers and Van Huylenbroeck (2007) and confirms the EE = TE x EAE decomposition (an internal-consistency check, not a replication of a printed table).
• Identity and property checks. The decompositions' multiplicative identities, GML = EC x BPC, score ranges and infeasibility semantics are asserted across randomised technologies.

The vignettes reproduce the published-table replications in the open: vignette("replication", "pgt").

Vignettes

• introduction: the core workflow on the synthetic steel panel and a rice nitrogen-balance example.
• models: the estimating linear programs, stated in full.
• replication: the published-result replications above.
• comparing-axioms: compare_models() across the axiom systems.
• multiple-pollutants: multi-pollutant technologies, DMU-specific coefficients and the staged decomposition.
• productivity: pgt_ml() productivity change and boot_pgt() inference.

References

• Ayres, R. U., & Kneese, A. V. (1969). Production, consumption, and externalities. American Economic Review, 59(3), 282-297.
• 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
• 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
• Dakpo, K. H., Jeanneaux, P., & Latruffe, L. (2016). Modelling pollution-generating technologies in performance benchmarking: Recent developments, limits and future prospects in the nonparametric framework. European Journal of Operational Research, 250(2), 347-359. doi:10.1016/j.ejor.2015.07.024
• 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
• Lauwers, L. (2009). Justifying the incorporation of the materials balance principle into frontier-based eco-efficiency models. Ecological Economics, 68(6), 1605-1614. doi:10.1016/j.ecolecon.2008.08.022
• 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
• 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
• 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