Weak G-Disposability in Emissions Analysis

• Weak G-disposability is an emissions technology that enforces the material balance principle by linking polluting inputs, desirable outputs, and emissions.
• It distinguishes itself from by-production and joint disposability by imposing a summation constraint that ensures physically consistent estimation of marginal abatement costs.
• Empirical evidence on U.S. coal-fired power plants shows that reducing electricity output is far less costly than reducing fuel inputs, yielding MAC estimates around $8.9k–$9.2k per 1000 tons CO2.

Searching arXiv for the specified paper and closely related work on weak G-disposability and material-balance-based emissions technologies. Weak G-disposability is an emission-generating technology used to estimate marginal abatement cost (MAC) under explicit physical constraints linking production, polluting inputs, and emissions. In "Quantile estimation of CO2 marginal abatement cost across emission-generating technologies" (Delnava et al., 16 Aug 2025), weak G-disposability is presented as a formulation consistent with the material balance principle (MBP), under which emissions are treated not as detached “bad outputs” but as residuals arising from the physical transformation of polluting inputs. Within that framework, abatement is necessarily costly because any reduction in emissions must occur through a physically consistent trade-off among emission-generating inputs, desirable output, and undesirable output.

1. Definition and material-balance foundation

The paper treats weak G-disposability as an emissions technology designed to satisfy the material balance principle. The MBP is defined as the requirement that emissions obey conservation-of-mass/energy logic, because they arise from the physical transformation of polluting inputs rather than from an abstract joint-production relation. The paper argues that ignoring MBP can produce MAC estimates that are economically misleading and physically inconsistent (Delnava et al., 16 Aug 2025).

Within this formulation, the technology is governed by convexity and by two essentiality conditions linking emissions and emission-generating inputs. The paper states that if $(x,y,b)\in \Ts_{WGD}$ and $b=0$, then $x^P=0$; and if $(x,y,b)\in \Ts_{WGD}$ and $x^P=0$, then $b=0$. It further imposes weak G-disposability through the condition that if $(x,y,b)\in \Ts_{WGD}$ and $u g_x + r g_y - g_b = 0$, then $(x^p+g_x,\, y-g_y,\, b+g_b)\in \Ts_{WGD}$. Here, $u$ is the emission factor for the polluting input and $b=0$0 is the recuperation factor for desirable output (Delnava et al., 16 Aug 2025).

Economically, these restrictions imply that pollution reduction is not freely disposable. Abatement must respect a summation constraint linking reductions in polluting inputs, reductions in desirable output, and changes in emissions. This makes weak G-disposability a physically disciplined representation of mitigation behavior rather than a purely axiomatic description of “good” and “bad” outputs.

2. Distinction from by-production and joint disposability

The paper compares weak G-disposability with by-production (BP) and joint disposability (JD), and the distinction is central to its interpretation of MAC. In BP, the production process is split into an economic technology for $b=0$1 and an environmental technology for $b=0$2. The paper’s critique is that conventional BP can fail to capture the full physical linkage between the two sub-technologies unless they are carefully unified (Delnava et al., 16 Aug 2025).

JD is defined more simply. It assumes free disposability for non-emission-generating inputs and desirable outputs, together with weak disposability between polluting inputs and emissions. In the paper’s formulation, this takes the form of a proportional contraction: $b=0$3 Under JD, emissions and polluting inputs move together proportionally.

Weak G-disposability differs in that it does not merely require proportional scaling between polluting inputs and emissions. Instead, it imposes the material-balance-type restriction

$b=0$4

This condition links input adjustment, desirable-output adjustment, and undesirable-output adjustment through a conservation law. The paper therefore characterizes WGD as more restrictive and more physically grounded than JD, while also noting that it is more flexible than conventional joint production of good and bad outputs. That flexibility is accompanied by stronger informational demands, because implementation requires factor information such as $b=0$5 and $b=0$6.

A common misconception is that WGD is simply another version of weak disposability. In the paper’s treatment, that is not the case. WGD is distinguished precisely by the summation constraint that embeds material balance directly into the technology.

3. Technology-set formulation

The weak G-disposability production possibility set is given explicitly as

$b=0$7

The paper interprets each component as follows. The condition $b=0$8 represents free disposability, in the usual weak sense, for non-emission-generating inputs. The equality $b=0$9 introduces slack-based adjustment for emission-generating inputs. The equality $x^P=0$0 analogously adjusts desirable output, and $x^P=0$1 adjusts undesirable output. The core WGD restriction is

$x^P=0$2

which the paper identifies as the material-balance or summing-up constraint. The condition $x^P=0$3 imposes variable returns to scale (Delnava et al., 16 Aug 2025).

The paper also states that, empirically, the slacks $x^P=0$4 replace directional vectors $x^P=0$5. In the implementation it follows Rodseth et al. by fixing direction vectors and setting $x^P=0$6 in the summing-up constraint to simplify estimation. This suggests that practical estimation of WGD often trades some generality for tractability while preserving the material-balance structure.

4. Estimation under full and quantile frontiers

For full-frontier estimation, the paper uses a sign-constrained CNLS model. The WGD estimator is defined by the objective

$x^P=0$7

subject to

$x^P=0$8

$x^P=0$9

$(x,y,b)\in \Ts_{WGD}$0

$(x,y,b)\in \Ts_{WGD}$1

$(x,y,b)\in \Ts_{WGD}$2

The paper identifies two features as distinctive. First, $(x,y,b)\in \Ts_{WGD}$3 is the WGD/material-balance condition in dual form. Second, $(x,y,b)\in \Ts_{WGD}$4 is the translation property. It also stresses that inputs are not freely disposable in this model, which differentiates WGD from BP and JD formulations (Delnava et al., 16 Aug 2025).

For quantile estimation, the paper replaces the squared-error CNLS objective with a convex expectile regression (CER) objective, producing a quantile frontier estimator. In the CER framework, residuals are split into positive and negative parts with asymmetric weighting. The stated purpose is to estimate the frontier at a chosen quantile $(x,y,b)\in \Ts_{WGD}$5, capture local inefficiency rather than imposing a single full frontier on all firms, and improve robustness to outliers and noise. For WGD specifically, the paper reports that the quantile estimator is less sensitive to inefficiency heterogeneity and empirically tends to perform better than the full frontier estimator.

The estimation framework therefore assigns WGD a dual role: it is both a technology specification and a set of identifying restrictions on shadow prices and trade-offs. This is important because MAC estimates in the paper are not generated by an unconstrained frontier, but by a frontier whose admissible movements must respect material balance.

5. Empirical implications for U.S. coal-fired power plants

The empirical application uses 71 bituminous coal-fired power plants in 2022. For weak G-disposability, the paper reports the following MAC values from Table 4: under the full frontier estimator, mean MAC is 8875.65 and median MAC is 9622.23; under the quantile frontier estimator, mean MAC is 9193.66 and median MAC is 9259.73 (Delnava et al., 16 Aug 2025).

The paper decomposes abatement cost into two components: $(x,y,b)\in \Ts_{WGD}$6, interpreted as the cost of reducing emissions through forgone electricity output, and $(x,y,b)\in \Ts_{WGD}$7, interpreted as the cost of reducing emissions through reduced fuel use. Under WGD, the full-frontier mean values are $(x,y,b)\in \Ts_{WGD}$8 and $(x,y,b)\in \Ts_{WGD}$9; the quantile-frontier mean values are $x^P=0$0 and $x^P=0$1. Because $x^P=0$2, the paper concludes that reducing electricity output is much cheaper than reducing fuel input.

This conclusion is stated strongly in the empirical discussion. Downscaling electricity production is more cost-effective than reducing fuel consumption. Under WGD, reducing fuel consumption is cost-ineffective for almost all plants, and only a negligible fraction of plants would prefer fuel reduction. The paper therefore interprets the least-cost abatement pathway under WGD as output reduction rather than input reduction.

The distributional pattern differs from BP and JD. Under WGD, mean MAC is below median MAC, whereas under BP and JD mean exceeds median. The paper attributes this to the restrictive WGD summation constraint, especially the setting $x^P=0$3, which effectively drives the dual price of desirable output $x^P=0$4 near zero. It further notes that, to avoid infinite MRT, $x^P=0$5 was replaced by $x^P=0$6. This indicates that the empirical shape of the MAC distribution is not only a property of the data but also a consequence of the WGD restriction set.

For plants at the lower and upper ends of the MAC distribution, the paper reports that the plants with the lowest MAC are very small-scale and use minimal fuel, while plants with the highest MAC are also relatively small-scale, though somewhat larger than the lowest-MAC plants. It interprets this as evidence that emission reduction is more feasible at lower coal consumption levels without large output losses.

6. Simulation evidence, robustness, and interpretive limits

The Monte Carlo analysis compares BP, JD, and WGD under two data-generating-process scenarios. The general finding is that WGD consistently yields the lowest RMSE across both scenarios, and CER consistently outperforms CNLS. The paper further states that WGD is especially strong when the DGP is unified and includes undesirable outputs directly (Delnava et al., 16 Aug 2025).

In Scenario 1, where desirable and undesirable outputs are modeled separately, WGD has lower RMSE than BP and JD for both CNLS and CER, and CER outperforms CNLS for all technologies. The paper also reports that higher inefficiency dispersion can lower RMSE, interpreting this as providing the estimator with a clearer signal. In Scenario 2, where desirable output depends jointly on inputs and undesirable output, WGD again performs best; its RMSE values are described as dramatically lower than those of BP and JD, and CER is especially accurate under WGD. The paper attributes this to WGD aligning more closely with a unified physical production process in which emissions directly affect output.

These simulation results support three broader claims made in the paper: WGD is the most physically coherent and statistically reliable technology among the three considered; quantile frontier estimation is more robust than full frontier estimation; and WGD is especially suitable when the data-generating process conforms to a joint production structure with material balance.

At the same time, the paper identifies interpretive limits. The WGD summing-up constraint is highly restrictive; the model requires DMU-specific knowledge of emission and recuperation factors $x^P=0$7; and the empirical implementation simplifies estimation by fixing direction vectors and setting $x^P=0$8. A plausible implication is that WGD’s advantages depend partly on the extent to which these physical coefficients are measured credibly and the production process is well represented by the MBP-based structure.

7. Conceptual significance for marginal abatement cost analysis

Within the paper’s comparative framework, weak G-disposability functions as the most physically disciplined representation of emissions technology. It combines essentiality conditions, material balance, and constrained joint adjustment of polluting inputs, desirable output, and undesirable output. Compared with by-production, it does not separate the economic and environmental mechanisms into loosely linked sub-technologies. Compared with joint disposability, it goes beyond proportional contraction by enforcing a conservation-based summation restriction (Delnava et al., 16 Aug 2025).

Its significance for MAC analysis lies in how technology assumptions alter the shadow-price interpretation of abatement. Under WGD, MAC is not simply the price of an undesirable output along an estimated frontier; it is the cost of moving within a technology set whose feasible adjustments are physically constrained. This is why the paper treats WGD as both an economic and a physical specification of abatement behavior.

The paper’s bottom-line synthesis is correspondingly specific. Weak G-disposability is presented as the most physically coherent emissions technology among the three considered; when paired with quantile frontier estimation, it produces more robust and accurate shadow prices; and in the empirical application to U.S. coal-fired power plants, it yields MAC estimates around \$x^P=0$99.2k per 1000 tons of CO$b=0$0 while strongly indicating that reducing electricity output is much cheaper than reducing fuel inputs.