--- title: "Introduction to pgt" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Introduction to pgt} %\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) ``` ## The materials-balance principle A pollution-generating technology turns material inputs into a good output and an unavoidable bad output. The pollutant content of the inputs has to end up somewhere: embodied in the product, removed by abatement, or emitted. `pgt` builds efficiency models on that accounting identity, $$u'x_l - v y_l \ge b_l,$$ where $x_l$ are the material inputs of unit $l$, $u$ their pollutant flow coefficients (for example tonnes of CO2 potential per unit of input), $y_l$ the good output, $v$ the pollutant retained in the product, and $b_l$ the emissions. The gap $u'x_l - v y_l - b_l$ is the pollutant that leaves the process other than through the measured emission. A note on terminology: the mass balance is an identity when every exit is measured; the inequality form above allows for unmeasured abatement, and when the abatement output $a_l$ is observed the account closes as the equality $u'x_l - v y_l = b_l + a_l$. Following Rødseth (2025), the documentation calls both forms the materials-balance identity. The principle itself is old: Ayres and Kneese (1969) built the materials-balance view of production and externalities, Lauwers (2009) made the case for carrying it into frontier-based eco-efficiency models, and Dakpo, Jeanneaux and Latruffe (2016) survey how pollution-generating technologies have been modelled in nonparametric benchmarking since. Existing R packages handle undesirable outputs by transforming the data (the Seiford and Zhu 2002 translation in `deaR`) or by imposing weak disposability and directional distances (`nonparaeff`, `Benchmarking`). Neither approach carries a materials-balance account. `pgt` audits the account before estimation, lets it enter the estimators as a cap, an equality, a price system or an objective valuation depending on the model, and puts the competing axiom systems behind one interface. ## Building a technology The constructor bundles the data with the flow coefficients. The shipped `steeldemo` data is a synthetic panel of steel plants on two production routes, with emissions in CO2 units, an observed abatement output (CO2 captured by carbon capture and storage, capture for utilisation, or slag mineralisation at the plants that run them) and the CO2 potential of the energy the capture consumes as a pollution-control input. ```{r} data(steeldemo) tech <- pgt_tech( x = steeldemo[, c("coal_coke", "other_fuel", "raw_material", "flux", "capture_energy")], y = steeldemo$production, b = steeldemo$emissions, a = steeldemo$captured, v = 0.01467, x_abate = "capture_energy", group = steeldemo$route, id = steeldemo$plant ) tech ``` The inputs already sit in pollutant-potential units, so the default $u = 1$ applies to each; `v = 0.01467` is the carbon retained per unit of crude steel (the synthetic generator's assumption; see `?steeldemo` for the derivation). `captured` fills the abatement slot `a`, and `x_abate` marks `capture_energy` as the pollution-control input, which the five-component decomposition below frees in its own stage while every estimator treats it like any other input. ## Auditing the identity before estimation `mb_check()` reports the closure gap per DMU and flags the accounts where more pollutant leaves than enters. The audit is the data-quality check behind the materials-balance interpretation of every model, and it gates the models that constrain the projection by the DMU's own account: the input-fixed benchmark `"wgd_input_fixed"` and the directional model `"fdmo"`. ```{r} mb <- mb_check(tech) head(as.data.frame(mb)) attr(mb, "n_violations") ``` With the abatement output observed, the audit also reports the closure of the full identity $u'x - v y = b + a$ (the `closure` column), which is exact on this panel: every tonne of CO2 potential is accounted for by the product, the emission or the captured stream. An infeasible `"wgd_input_fixed"` programme always belongs to a DMU with a strictly negative gap, so the audit locates any estimation problem in advance; such DMUs return `NA` scores and `pgt()` warns. The weak-G-disposability programme of Eq. 6 is always self-feasible, so its scores lie in $(0, 1]$ on any data. ## Estimating environmental efficiency The weak-G-disposability model of Rødseth (2025, Eq. 6) minimises emissions over peer mixes that hold the good output at the DMU's level, pricing any extra good output at the DMU's retained-content coefficient $v$; the inputs are decision variables (the programmes are stated in full in `vignette("models", "pgt")`). The score is $b^*_l / b_l$. ```{r} fit <- pgt(tech, model = "wgd") summary(fit) ``` The envelope model drops the retained-content term (the $v = 0$ case) and reduces the programme to the convex lower envelope of the $(y, b)$ scatter. ```{r} fit_env <- pgt(tech, model = "envelope") summary(fit_env) ``` The directional representation of Rødseth (2025, Eq. 14) needs the closed account and holds abatement at the DMU's own level; it expands the good output and contracts the bad output jointly, so its scores are in output units rather than ratios, and the materials-balance row ties the two together as $\theta_b = v\,\theta_y$. ```{r} fit_dir <- pgt(tech, model = "fdmo") summary(fit_dir) ``` ## Shadow prices and abatement cost The output-constraint dual $\eta_l = \partial b^*/\partial y$ (returned as `dual_output`) measures the marginal emission content of the good output along the frontier. `mac_curve()` turns it into a marginal abatement cost curve: reducing the bad output by one unit costs $1 / \eta_l$ units of the good output, or $p / \eta_l$ at output price $p$. Read the curve with care about which margin is which: each DMU's plotted abatement is its distance to the frontier, $b_l - b^*_l$, which the model itself prices at zero output loss, while the `mac` value is the marginal cost of abating beyond the DMU's frontier point. The curve therefore ranks DMUs by the shadow price at their projection; the area under it is not a total-cost estimate. ```{r, fig.alt = "Step curve of marginal abatement cost against cumulative abatement potential"} head(shadow_prices(fit)) mac <- mac_curve(fit, price = 550) plot(mac) ``` ## Decomposing across production routes With a technology group, `pgt_decompose()` splits environmental efficiency into a within-group and a technology-gap component. The envelope decomposition is an exact identity, Total $=$ WR $\times$ TGR, where WR is within-route reallocation and TGR the technology-gap ratio of the metafrontier tradition (Battese, Rao and O'Donnell 2004; O'Donnell, Rao and Battese 2008). ```{r} dec <- pgt_decompose(tech, type = "envelope") summary(dec) ``` The five-component source decomposition of Rødseth (2025, Eq. 11) traces each score to production technical efficiency, coefficient quality, production allocative efficiency, abatement technical efficiency and abatement allocative efficiency; it needs no groups, and the stage programmes are stated in `vignette("models", "pgt")`. The abatement components have something to measure here because the panel observes `captured` and marks `capture_energy` as the control input. ```{r} dec5 <- pgt_decompose(tech, type = "rodseth") summary(dec5) comps <- c("te_production", "quality", "ae_production", "te_abatement", "ae_abatement", "total") med <- aggregate(dec5$results[comps], list(abatement = steeldemo$abatement_tech), median) med[comps] <- round(med[comps], 3) med ``` Two features of the table follow from the stage programmes. Quality is 1 throughout because the coefficients are homogeneous. Production technical efficiency is 1 at the plants without capture and below 1 at many capturing plants: the first stage holds the captured tonnage at the plant's own level only as a floor, so a mix of peers that captures more from the same inputs reaches a lower controlled emission. Production allocative efficiency, the stage that frees the material inputs, carries most of the gap. The abatement components separate the plants by what they capture: abatement technical efficiency falls below 1 only where capture is active, where it measures how far the plant's captured tonnage falls short of the best capture observed at no more capture energy, and abatement allocative efficiency, the stage that frees the capture-energy input, is lowest at the plants that capture nothing, where it measures the emission reduction available from adopting capture on the peers' terms. ## A worked example on real data The `frontier` package ships a balanced panel of Philippine rice farms with a physical fertiliser input (`NPK`, kilograms of active ingredient) and a physical output (`PROD`, tonnes of paddy). Treating the nitrogen carried by the fertiliser as the pollutant potential gives a nitrogen materials balance. The nitrogen fraction below is illustrative (urea-equivalent, about 0.46 of the active ingredient). ```{r, eval = requireNamespace("frontier", quietly = TRUE)} data("riceProdPhil", package = "frontier") d8 <- riceProdPhil[riceProdPhil$YEARDUM == 8, ] uN <- 0.46 rice <- pgt_tech( x = as.matrix(d8[, c("AREA", "LABOR", "NPK", "OTHER")]), y = d8$PROD, b = uN * d8$NPK, u = c(AREA = 0, LABOR = 0, NPK = uN, OTHER = 0), v = 0, id = as.character(d8$FMERCODE) ) rice_fit <- pgt(rice, model = "wgd") summary(rice_fit) ``` Here environmental efficiency is the scope to cut fertiliser nitrogen while holding rice output at its observed level. One caveat on this construction: because the bad output is defined as the full nitrogen potential of a single input ($b = u_N \cdot$ `NPK`), every farm's account closes with equality by construction and `mb_check()` passes trivially; the example illustrates the workflow, not the audit. The other vignettes state the estimating programmes in full (`vignette("models", "pgt")`), reproduce the published-result replications (`vignette("replication", "pgt")`), compare the axiom systems (`vignette("comparing-axioms", "pgt")`), demonstrate multi-pollutant technologies and heterogeneous coefficients (`vignette("multiple-pollutants", "pgt")`) and cover productivity measurement with inference (`vignette("productivity", "pgt")`). ## 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 - 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 - 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 - 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 - 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 - Seiford, L. M., & Zhu, J. (2002). Modeling undesirable factors in efficiency evaluation. *European Journal of Operational Research*, 142(1), 16-20. doi:10.1016/S0377-2217(01)00293-4