📈 economics

A By-Production Approach to Deriving the Production Possibility Frontier and the Shadow Prices of CO₂ Reduction

This paper proposes a novel by-production optimization framework to derive an explicit Production Possibility Frontier and unique shadow prices for CO₂ reduction, enabling the direct estimation of marginal abatement costs as opportunity costs without relying on arbitrary directional vectors.

Original authors: Alexandre Repkine

Published 2026-07-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexandre Repkine

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the economy as a giant, bustling kitchen. In this kitchen, chefs (countries) use ingredients like capital, labor, and fuel to cook up two very different dishes: a delicious, desirable meal (GDP) and a smelly, unwanted cloud of smoke (CO₂ emissions). Usually, when you want less smoke, you just turn down the stove. But here's the twist: in this kitchen, the smoke isn't just a side effect; it's baked into the recipe. You can't make the meal without making some smoke, and turning down the smoke often means serving a smaller meal.

This paper by Alexandre Repkine introduces a new, crystal-clear way to map out exactly how much delicious food you have to give up to get a little less smoke. The author calls this map the Production Possibility Frontier (PPF). Think of it as the "edge of the kitchen" where the chefs are working at their absolute best.

The Problem with Old Maps

Before this paper, economists tried to draw this map using a method that was a bit like guessing the slope of a hill by throwing a ball in a random direction. They used something called "directional distance functions," which required picking an arbitrary direction to measure the trade-off. The paper argues this is messy and unreliable. It's like trying to measure how steep a mountain is by walking in a direction you just guessed, rather than looking at the actual cliff face. The author says these old methods can give weird results, especially if you are already standing right on the edge of the efficient zone.

The New "By-Production" Recipe

Instead of guessing directions, this paper uses a "by-production" approach. It treats the kitchen as having two separate but linked recipes happening at once: one for the good meal and one for the smoke. The author sets up a strict math puzzle (a constrained optimization problem) to find the absolute maximum amount of food you can cook for any specific limit on smoke.

The paper proves that if you follow certain rules about how the kitchen works (like ensuring the ingredients behave nicely and the smoke doesn't magically disappear), you can draw a single, smooth, unique line that represents the best possible trade-off. This line is the Production Possibility Frontier.

The Price of Clean Air

Once you have this line, the "price" of cleaning up the air is simply the slope of the line. If the line is steep, you have to give up a lot of food to get a tiny bit less smoke. If it's flat, it's cheaper to clean up. The paper calculates this slope directly, calling it the Marginal Abatement Cost (MAC).

The author applied this method to data from 16 Asia-Pacific economies between 2010 and 2019. They didn't just guess; they used real numbers for GDP, capital, labor, and fuel consumption to estimate the shape of this frontier.

What the Numbers Say

The results are striking. The paper found that the cost of reducing pollution depends heavily on how much pollution a country is already making.

China, the biggest polluter in the group with an average of 10.717 gigatons of CO₂, has the lowest cost to clean up: about $1,727 per ton.
Sri Lanka, a much smaller polluter with only 0.02 gigatons of CO₂, faces a much steeper hill. To reduce one ton of CO₂, they would have to sacrifice about $10,555 in economic output.

The paper suggests that for smaller economies, the trade-off is incredibly steep. It's like trying to squeeze the last drop of juice out of a nearly empty orange; it takes a lot of effort for very little result.

What This Is Not

It is crucial to understand what this paper is not measuring. The author explicitly states that these numbers represent the cost of reducing emissions without changing the technology. It's the cost of simply turning down the stove right now, using the same old recipes.

The paper argues against the idea that these high numbers mean we can't clean up. Instead, it suggests that these high costs highlight the massive value of inventing new recipes (technological change). Other studies that show much lower costs (like $40 or $60 per ton) are measuring the cost of switching to new, cleaner fuels or better technology. This paper measures the cost of the "painful squeeze" if we don't change the technology.

The Takeaway

The paper concludes that while the math is complex, the message is clear: we need to stop guessing the direction of the trade-off and start looking at the actual shape of the frontier. By showing exactly how steep the hill is for different countries, this new method gives policymakers a better map. It shows that for some countries, the cost of cleaning up without new technology is enormous, suggesting that the real solution lies in innovation and new technologies, not just in squeezing the existing system harder.

The author admits that this is a theoretical framework applied to real data, and while the math proves the map exists and is unique under certain conditions, the real-world application involves assumptions about how countries behave. However, the method offers a fresh, direct way to see the true economic trade-off between a growing economy and a clean sky.

Technical Summary: A By-Production Approach to Deriving the Production Possibility Frontier and the Shadow Prices of CO₂ Reduction

Problem Statement
Existing literature on estimating Marginal Abatement Costs (MACs) for harmful emissions relies heavily on Directional Output Distance Functions (DODFs), both parametric and non-parametric (DEA). While theoretically grounded, these methods face significant limitations:

Sensitivity and Arbitrariness: DODF-based estimates are highly sensitive to the choice of directional vectors, which are often selected arbitrarily (e.g., 45-degree vectors) without economic rationale.
Undefined MACs: In non-parametric DEA approaches, MACs (defined as the slope of the Production Possibility Frontier, or PPF) can become undefined at the vertices of the piecewise-linear efficiency frontier.
Marginal vs. Non-Marginal: Standard approaches are inherently marginal, often failing to approximate the economic costs of large-scale, non-marginal emission reductions.
Implicit vs. Explicit: Most studies do not explicitly estimate the functional form of the PPF linking desirable output ( $y$ ) and undesirable output ( $b$ ). Instead, they derive MACs indirectly as ratios of derivatives of distance functions at a projection point, rather than as the slope of an explicit PPF function.

Methodology
The paper proposes a novel framework to derive an explicit Production Possibility Frontier (PPF) and corresponding MACs within the by-production framework originally developed by Murty et al. (2012).

Theoretical Framework:
- The polluting technology is modeled as the intersection of two sub-technologies: one producing the desirable output ( $y$ ) and one producing the undesirable output ( $b$ ), linked by the use of polluting inputs (specifically fuel, $f$ ).
- The authors formulate a constrained optimization problem to maximize desirable output $y$ $y$ subject to:
  - Feasibility constraints defined by two distance functions, $D_y(y, k, l, f) \leq 1$ and $D_b(b, k, l, f) \leq 1$ .
  - An exogenous emissions cap $b \leq b_0$ .
- Regularity Conditions: To ensure the PPF is well-behaved (unique, existing, and efficient), the authors impose four specific conditions on the distance functions:
  1. Strict monotonicity of $D_y$ in $y$ and $D_b$ in $b$ .
  2. Continuous differentiability of both functions.
  3. Compactness and non-emptiness of the feasible set.
  4. A non-degeneracy condition ensuring the gradient of $D_y$ is non-zero in the interior.
- Theorem 1: Under these conditions, the authors prove the existence and uniqueness of the PPF, denoted $y^*(b_0)$ . They demonstrate that at the optimum, both distance function constraints bind, and the emissions constraint binds ( $b^* = b_0$ ).
- MAC Definition: The Marginal Abatement Cost is derived directly as the slope of this explicit PPF ( $dy^*/db$ ), which corresponds to the Lagrange multiplier of the emissions constraint. This aligns MACs with the classical definition of the Marginal Rate of Transformation.
Functional Specification and Estimation:
- The authors specify Cobb-Douglas functional forms for the distance functions:
  - $D_y = y / (C k^\alpha l^\beta f^\gamma)$
  - $D_b = (T k^{\theta_1} l^{\theta_2} f^{\theta_3}) / b$
- These specifications satisfy the null-jointness property (zero good output implies zero bad output).
- Endogenous Proportionality: Solving the first-order conditions of the Lagrangian reveals a necessary condition where the ratios of output elasticities to input elasticities must be equal across inputs ( $\alpha/\theta_1 = \beta/\theta_2 = \gamma/\theta_3 = m$ ). This parameter $m$ defines the shape of the PPF.
- Explicit PPF Derivation: The solution yields an explicit power-function PPF: $y^*(b_0) = (C/T) \cdot b_0^m$ .
- Estimation Strategy: The parameters are estimated using Stochastic Frontier Analysis (SFA) on a balanced panel of 16 Asia-Pacific economies (2010–2019). The model accounts for random noise and technical inefficiency in both the good and bad output production processes.

Key Results

Empirical Application: Using data from 16 Asia-Pacific economies (including China, India, Japan, and Australia), the authors estimated the parameters of the polluting technology.
PPF Shape: The estimated parameter $m$ was 0.715. Since $m < 1$ , the resulting PPF is concave, consistent with the economic intuition that marginal abatement costs increase as emissions decrease (or conversely, that reducing emissions becomes cheaper as the emission level rises).
MAC Estimates:
- MACs were calculated as the slope of the PPF: $MAC(b_0) = m(C/T) b_0^{m-1}$ .
- Results show a strong inverse relationship between the level of emissions and MACs. Large emitters like China (10.7 Gt CO₂) have low MACs (~~$1,727/ton), while smaller emitters like Sri Lanka (0.02 Gt CO₂) have high MACs (~~$10,555/ton).
- The estimates are substantially higher (by orders of magnitude) than recent studies (e.g., Jiang et al., 2022; Chu, 2023) that report MACs in the range of $40–$ 300.

Significance and Claims
The paper claims three primary contributions:

Explicit Derivation: It provides a method to derive an explicit functional form of the PPF directly from the underlying by-production technology, rather than inferring it implicitly from distance function derivatives.
Theoretical Rigor: It establishes precise, relatively mild regularity conditions (beyond those in Murty et al., 2012) that guarantee the existence, uniqueness, and efficiency properties of the PPF, ensuring MACs are well-defined.
Policy Relevance: By deriving MACs as the slope of an explicit PPF, the method avoids the arbitrary choice of directional vectors required in DODF approaches. Furthermore, because it models the entire frontier, it allows policymakers to evaluate the economic costs of non-marginal (large-scale) emission reductions, rather than just infinitesimal changes.

Interpretation of High MAC Estimates
The authors explicitly address the discrepancy between their high MAC estimates and lower values in other literature. They argue that their estimates represent the pure opportunity cost of reducing emissions while holding technology constant (moving along the existing PPF). In contrast, other studies often capture the cost of shifting the PPF outward through technological change or input substitution (e.g., fuel switching). Thus, the high values in this study highlight the significant economic sacrifice required to reduce emissions without technological progress, underscoring the value of innovation.

Limitations and Future Directions
The paper acknowledges several limitations and areas for future research:

Endogeneity of Inefficiency: The model assumes independent inefficiency terms for good and bad outputs, though these may be correlated.
Functional Form: The theoretical basis for applying Cobb-Douglas forms to the bad-output distance function is less firm than for the good-output function.
Technological Heterogeneity: The current analysis assumes a single global frontier. Future work could employ a meta-frontier framework to account for differences in technology across country groups.
Concavity Conditions: While the empirical PPF is concave, the paper notes that global strict convexity of the feasible set (a sufficient condition for PPF concavity) is too restrictive for empirical reality. Identifying milder necessary and sufficient conditions for PPF concavity remains an open task.