Joint Disposability in Production Theory

• Joint disposability is an environmental production theory concept modeling technologies that jointly produce desirable outputs and pollution from non-emission and emission-generating inputs.
• It enforces free disposability for non-emission inputs and desirable outputs while imposing weak disposability on polluting inputs and emissions, linking CO2 generation to fuel use.
• Empirical analysis for US coal-fired plants shows that under joint disposability, reducing output is often the least-cost abatement channel, especially with quantile frontier estimation methods.

Searching arXiv for papers on “joint disposability” and closely related terminology. Joint disposability is an environmental production-theory concept used to model technologies that jointly generate desirable output and undesirable output from heterogeneous inputs. In the formulation studied in "Quantile estimation of CO2 marginal abatement cost across emission-generating technologies" (Delnava et al., 16 Aug 2025), firms use non-emission-generating inputs $x^N$ and emission-generating inputs $x^P$ to produce desirable outputs $y$ and undesirable outputs $b$, and the defining feature of joint disposability is that non-emission-generating inputs and desirable outputs are freely disposable while emissions and emission-generating inputs are weakly disposable together. In that paper, joint disposability (JD) is analyzed alongside by-production (BP) and weak G-disposability (WGD) as an alternative emission-generating technology for estimating CO$_2$ marginal abatement cost (MAC) for U.S. coal-fired power plants, and the technology choice materially affects both the inferred abatement mechanism and the estimated MAC levels (Delnava et al., 16 Aug 2025).

1. Conceptual meaning and analytical role

Within the paper’s environmental production framework, JD models production when firms use non-emission-generating inputs $x^N$, emission-generating inputs $x^P$, produce desirable outputs $y$, and also generate undesirable outputs $b$ such as CO$_2$ (Delnava et al., 16 Aug 2025). The central idea is that emissions are not freely disposable: reducing them requires reducing the associated polluting input, and in practice this can also constrain desirable output. The paper therefore treats JD as a technology in which emissions remain tied to polluting material or energy use rather than as a byproduct that can be independently “thrown away.”

The paper identifies three relations that the technology is intended to preserve: the positive relation between polluting inputs and emissions, the positive relation between inputs and electricity generation, and the positive relation between desirable and undesirable outputs (Delnava et al., 16 Aug 2025). In the empirical application, $x^P$0 is net electricity generation, $x^P$1 is CO$x^P$2 emissions, $x^P$3 is total fuel consumption, and $x^P$4 is plant nameplate capacity and operating availability. This formulation is used because CO$x^P$5 from coal-fired generation is physically linked to fossil-fuel use, and the paper treats JD as a plausible representation when emissions are tied closely to fuel combustion and end-of-pipe CO$x^P$6 treatment options are limited in the data.

A further conceptual role of JD in the paper is its connection to the material balance principle (MBP). The authors compare BP, JD, and WGD because all three are presented as consistent with MBP, in contrast to older models that may violate thermodynamic conservation (Delnava et al., 16 Aug 2025). For JD, that consistency is indirect: emissions are linked to emission-generating inputs, and weak disposability of $x^P$7 prevents emissions from being reduced independently of the polluting flow.

2. Formal axioms and exact technology representation

The paper partitions inputs as

$x^P$8

where $x^P$9 denotes $y$0 non-emission-generating inputs and $y$1 denotes $y$2 emission-generating inputs (Delnava et al., 16 Aug 2025). JD is then defined by three assumptions.

The first is free disposability of inputs and desirable outputs: $y$3

The second is weak disposability between emissions and emission-generating inputs: $y$4

The third is the JD technology itself: $y$5

The paper’s central mathematical definition is the consolidated DEA/VRS representation

$y$6

This representation gives nonpolluting inputs and desirable outputs the usual free-disposal inequalities, while imposing equalities for polluting inputs and undesirable outputs. The equalities

$y$7

are the formal expression of jointness in the paper: polluting inputs and bad outputs are benchmarked together and are not separately disposable (Delnava et al., 16 Aug 2025).

The paper explicitly rejects a decentralized JD approach with two intensity vectors and adopts a consolidated, single-$y$8 formulation. Its rationale is stated directly: “Recognizing the interdependence of processes like electricity generation and pollution emission, the specification of two separate intensity variables is questionable. Therefore, benchmark bundles are created by treating the entire input-output bundle as a single peer group vector” (Delnava et al., 16 Aug 2025).

3. Position relative to by-production and weak G-disposability

JD is presented comparatively, not in isolation. BP separates production into an economic sub-technology $y$9 and an environmental sub-technology $b$0, so that

$b$1

with separate intensity vectors $b$2 and $b$3. WGD instead uses one intensity vector plus slack variables and imposes the summing-up condition

$b$4

JD occupies an intermediate position: it uses one intensity vector like an integrated technology, but unlike WGD it has no explicit summing-up equality; unlike BP it does not decompose the process into two sub-technologies (Delnava et al., 16 Aug 2025).

Technology	Structure	Distinguishing restriction
BP	$b$5 with $b$6	Separate economic and environmental sub-technologies
JD	$b$7 with one $b$8	$b$9, $_2$0
WGD	One $_2$1 plus slacks	$_2$2

The paper characterizes these differences precisely. Relative to BP, JD uses one integrated technology and one common benchmark bundle instead of separate convex combinations for the economic and residual sides (Delnava et al., 16 Aug 2025). Relative to WGD, JD does not impose the directional or slack-based conservation relation

$_2$3

or, empirically,

$_2$4

The paper therefore describes JD as more structured than simple free disposability, less physically explicit than WGD, and less decomposed than BP.

This comparative placement matters for interpretation. BP is especially suited to representing production and pollution generation as related but conceptually distinct mechanisms. WGD is more restrictive and explicitly grounded in a directional or slack-based material-balance summation constraint. JD instead preserves the dependence of emissions on polluting inputs without requiring the explicit accounting identity used by WGD (Delnava et al., 16 Aug 2025).

4. Abatement channels and marginal abatement cost

Under JD, the paper interprets emissions abatement as proceeding through three channels: lowering desirable output $_2$5, lowering emission-generating inputs $_2$6, and moving to the frontier by eliminating inefficiency (Delnava et al., 16 Aug 2025). This means that abatement is not modeled as an independent end-of-pipe disposal channel. The economic intuition given is that if a plant wants to lower emissions, it may need to burn less fuel, and that often means producing less electricity unless efficiency improves.

The paper writes the JD directional distance model as

$_2$7

subject to the VRS and directional constraints on $_2$8, $_2$9, $x^N$0, and $x^N$1. Despite typographical issues noted in the detailed exposition, the intended interpretation is explicit: an inefficient plant can be projected by increasing good output, reducing bad output, and reducing polluting input while keeping nonpolluting inputs fixed (Delnava et al., 16 Aug 2025).

For MAC, the paper first recalls the directional-distance marginal rate of transformation

$x^N$2

It then argues that conventional MRT-based MAC is too narrow because firms may also reduce emissions through input-side changes, and adopts

$x^N$3

where

$x^N$4

This formula gives JD a two-channel MAC interpretation. The first channel is output-side abatement, represented by $x^N$5, and the second is input-side abatement, represented by $x^N$6 (Delnava et al., 16 Aug 2025). The least-cost MAC is the smaller of these two channels. In the paper’s empirical reading, JD implies that reducing electricity output is usually much cheaper than reducing fossil-fuel inputs.

The dual sign-constrained CNLS representation used for JD is

$x^N$7

subject to

$x^N$8

$x^N$9

$x^P$0

$x^P$1

$x^P$2

In this system, $x^P$3 is the dual weight for desirable output, $x^P$4 for emission-generating input, $x^P$5 for emissions, and $x^P$6 for non-emission-generating inputs (Delnava et al., 16 Aug 2025). MAC-related quantities are then read from local frontier slopes, with $x^P$7 linked to $x^P$8 and $x^P$9 linked to $y$0, up to the directional-vector normalization.

5. Estimation strategies and empirical findings

The paper estimates JD with two strategies: full frontier estimation via sign-constrained CNLS and quantile frontier estimation via convex expectile regression (CER), used as an indirect quantile estimator (Delnava et al., 16 Aug 2025). The motivation is methodological as well as empirical: full frontier methods treat all deviations from the estimated frontier as inefficiency and can overestimate shadow prices because of inefficiency, limited abatement options, and data noise or outliers.

The paper defines the conditional quantile function as

$y$1

and the directional-distance representation for a $y$2-quantile frontier as

$y$3

The generic CER objective is

$y$4

subject to the JD shape constraints and

$y$5

with

$y$6

For JD, the paper considers

$y$7

Plants outside the range use the nearest quantile, and plants between two quantiles use the mean of adjacent estimates (Delnava et al., 16 Aug 2025). The variables are normalized, and the JD direction vectors are specified as

$y$8

In the empirical implementation, the radial MSE method yields

$y$9

For U.S. bituminous coal-fired power plants in 2022, JD produces the following MAC estimates (Delnava et al., 16 Aug 2025):

Estimator	MAC mean / median	$b$0 mean / median	$b$1 mean / median
Full frontier	$b$2	$b$3	$b$4
Quantile frontier	$b$5	$b$6	$b$7

These values show two features that the paper emphasizes. First, JD-based MAC falls sharply when moving from full frontier to quantile frontier estimation. Second, under JD, $b$8 is typically much smaller than $b$9, so the least-cost MAC almost always comes from the output-side tradeoff rather than from fuel reduction (Delnava et al., 16 Aug 2025). The paper states that under JD only $_2$0 of plants prefer reducing fuel use, whereas under BP about $_2$1 do and under WGD almost none do.

The JD distribution is described as positively skewed because mean MAC exceeds median MAC under both estimators (Delnava et al., 16 Aug 2025). The paper’s plant-level interpretation is also specific: under JD, plants with the lowest MAC operate at a very small scale, use minimal fuel inputs around nine times less than the sample average, and have correspondingly low emissions and low electricity output. Plants with the highest MAC also operate below average scale, with fuel consumption roughly two times below the average under JD. The lowest-MAC JD plant lies above the $_2$2th quantile, while the highest-MAC JD plant lies below the $_2$3th quantile, specifically between the $_2$4th and $_2$5th quantiles.

The Monte Carlo analysis is not exclusive to JD, but it is used to evaluate estimator performance under BP, JD, and WGD. For JD, Scenario 1 reports CNLS RMSE values of $_2$6, $_2$7, and $_2$8 at $_2$9, $x^P$00, and $x^P$01, while CER at $x^P$02 reports $x^P$03, $x^P$04, and $x^P$05 (Delnava et al., 16 Aug 2025). Scenario 2 reports CNLS values of $x^P$06, $x^P$07, and $x^P$08, and CER at $x^P$09 reports $x^P$10, $x^P$11, and $x^P$12. The detailed discussion notes that CER often improves on CNLS, though not uniformly at every quantile and every $x^P$13, while the paper’s overall interpretation is that the quantile estimator consistently delivers more accurate results than the full frontier estimator.

6. Scope, limitations, and terminological boundaries

The paper treats JD as a legitimate MBP-consistent technology and as more realistic than models that allow undesirable outputs to be freely disposable, but it does not present JD as universally dominant (Delnava et al., 16 Aug 2025). Its conclusions are explicitly comparative: JD yields much lower MAC than BP, usually indicates that output reduction is cheaper than fuel reduction, and is empirically plausible when emissions are tightly tied to fuel combustion and firms have limited abatement options beyond reducing activity or polluting input use.

At the same time, the paper implies several limitations. JD mainly captures output reduction, polluting input reduction, and efficiency improvement. It does not explicitly represent fuel switching, carbon capture, permit trading behavior within the technology itself, or other engineered abatement technologies (Delnava et al., 16 Aug 2025). It is also less physically explicit than WGD because it does not impose the conservation relation

$x^P$14

The paper further suggests that JD can misrepresent abatement opportunities if the technology is too restrictive or too stylized, because tying $x^P$15 and $x^P$16 closely may overstate the dependence of emissions on fossil-fuel contraction when other abatement technologies exist but are unobserved.

The expression “joint disposability” also requires careful terminological separation from unrelated uses of disposal language. In "Waste Makes Haste: Bounded Time Protocols for Envy-Free Cake Cutting with Free Disposal" (Segal-Halevi et al., 2015), the relevant notion is not joint disposability in production theory but “free disposal,” meaning that some parts of the cake may be left unallocated: $x^P$17 That paper studies a disposal assumption in cake cutting, not a technology set with jointly disposable inputs and outputs. Its “free disposal” is therefore a permitted nonallocation of residue rather than a production-theoretic disposability axiom.

A different terminological neighborhood appears in "Post-processing minimal joint observables" (Heinosaari et al., 2018). There the relevant notions are compatibility, joint observability, and minimality in the post-processing preorder for finite-outcome POVMs, not disposability. The paper studies whether a family of compatible observables admits joint observables that are minimal under classical post-processing, and proves that every finite-outcome joint observable is lower bounded by a minimal joint observable in the post-processing order (Heinosaari et al., 2018). This is a problem in quantum measurement theory rather than in environmental production analysis.

In its most precise use here, joint disposability denotes the environmental production technology defined by free disposability of $x^P$18, weak disposability of $x^P$19, and the integrated benchmark technology

$x^P$20

with a single intensity vector (Delnava et al., 16 Aug 2025). Within that framework, the article’s central empirical implication is equally specific: for U.S. coal-fired power plants, JD implies that reducing electricity output is usually much cheaper than reducing fossil-fuel inputs as a means of lowering CO$x^P$21, and quantile-frontier estimation substantially lowers estimated MAC relative to full-frontier methods.