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An Order-Invariant Decomposition of Input Accumulation in DEA–Malmquist Productivity Analysis

This paper proposes a path-symmetrized decomposition method that resolves the order-dependence problem in multi-input DEA–Malmquist analysis, providing unique, order-invariant, and multiplicatively consistent measures of individual input accumulation effects.

Original authors: Kozo Kunimune

Published 2026-10-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kozo Kunimune

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

Economists have long sought to understand why some nations grow richer and more productive than others. To do this, they often look at a country's output—what it produces—and try to figure out how much of that growth comes from working harder, using better tools, or simply becoming more efficient at using what it already has. A standard tool for this kind of investigation is a method called Data Envelopment Analysis, which acts like a map of the most efficient possible performance a group of countries can achieve at any given time. This map, known as the "frontier," shows the best possible output for a specific amount of input, such as labor or capital. When a country moves closer to this frontier, it is catching up; when the frontier itself moves outward, it means technology has advanced. However, there is a third way productivity grows: by simply accumulating more inputs, like building more factories or training more workers. This is often described as moving along the existing frontier to a point where more resources are being used to create more output.

The challenge arises when a country uses more than one type of input, such as both physical machines and educated workers. If both of these increase at the same time, it becomes difficult to say exactly how much of the growth came from the machines and how much came from the education. The problem is that the answer can change depending on which factor you look at first. If you imagine adding the machines first and then the workers, you get one result. If you imagine adding the workers first and then the machines, you get a different result. This creates a confusing ambiguity where the story of economic growth depends entirely on the arbitrary order in which a researcher chooses to examine the factors.

Kozo Kunimune, a researcher at Kwansei Gakuin University, has proposed a solution to this puzzle. He developed a new way to break down the growth that comes from accumulating inputs so that the result no longer depends on the order in which the factors are considered. Instead of picking one sequence to follow, his method looks at every possible sequence in which the inputs could be added. It then takes an average of all these different paths to find a single, unique answer for how much each input contributed. This approach ensures that the final numbers are fair and consistent, regardless of how the calculation is arranged.

To test this idea, Kunimune applied it to real-world data from the Penn World Table, a major database of international economic statistics. He focused on the period between 2000 and 2010, looking at sixty-four countries that had complete data. In his analysis, he treated physical capital (machines and buildings) and human capital (the skills and education of the workforce) as two separate inputs. By using his new method, he was able to separate the growth in labor productivity into three distinct parts: improvements in how efficiently countries used their resources, shifts in the technological frontier, and the specific contributions of adding more physical capital and more human capital.

The results revealed that the way we measure economic growth can change significantly depending on how we model the inputs. When human capital was ignored and only physical capital was considered, the growth appeared to be driven largely by technological shifts. However, when human capital was included as a separate input, a portion of that growth previously attributed to technology was reclassified as movement along the frontier. In other words, some of what looked like a technological leap was actually the result of countries investing in their workers' skills. This shift was particularly noticeable in developing nations like Cambodia, Bangladesh, and Vietnam, where the inclusion of human capital as a distinct factor reduced the measured technological change and increased the measured accumulation of inputs.

The study also highlighted differences between advanced and emerging economies. In many wealthy nations, such as the United States, Canada, and Australia, the growth in productivity was driven almost entirely by the accumulation of physical capital, with human capital playing a minimal role in the measured expansion. In contrast, several emerging economies showed significant growth driven by the accumulation of human capital, suggesting that investments in education and skills were a major engine of their economic progress.

By removing the arbitrary element of ordering, this new method provides a clearer and more stable picture of what drives economic growth. It allows researchers to see the internal structure of productivity changes without the distortion caused by the order of calculation. The findings confirm that while the total amount of growth remains the same, the story behind that growth changes when we account for the specific roles of different inputs. This clarity is essential for policymakers and economists who need to understand whether a country's success is due to better technology, more efficient management, or the steady accumulation of physical and human resources. The method is designed to be flexible and can be applied to various economic settings, offering a transparent way to examine the complex machinery of productivity in a multi-input world.

Technical Summary: An Order-Invariant Decomposition of Input Accumulation in DEA–Malmquist Productivity Analysis

Problem Statement
In Data Envelopment Analysis (DEA)–Malmquist productivity analysis, total factor productivity change is typically decomposed into efficiency change (catch-up), technological change (frontier shift), and input accumulation (movement along the frontier). While the aggregate accumulation term is well-defined, a significant methodological difficulty arises in multi-input settings when attempting to attribute this accumulation to specific inputs (e.g., physical capital vs. human capital).

The core problem is order dependence. When multiple inputs change simultaneously, the contribution attributed to each individual input depends on the sequence in which those changes are evaluated along the productivity frontier. Different permutations of input changes yield different decompositions, rendering input-specific accumulation measures non-unique and incomparable across studies. While the literature has addressed evaluation-point dependence (e.g., via Fisher-ideal symmetrization for aggregate terms), it has largely overlooked this permutation dependence in genuinely multi-input technologies. Existing approaches, such as the "HR-series" (Henderson and Russell, 2005), often incorporate human capital as a labor-augmenting factor rather than an independent input, thereby sidestepping the multi-input attribution problem rather than solving it.

Methodology
The paper proposes a path-symmetrized decomposition to resolve this order dependence. The methodology treats the attribution of accumulation to individual inputs as a problem of averaging marginal contributions over all admissible paths, analogous to the Shapley value in cooperative game theory.

Counterfactual Construction: For an input vector moving from $x_t$ to $x_{t+1}$ , the method constructs counterfactual input vectors that represent intermediate states where only one input component has changed while others remain at their initial or final levels.
Path Symmetrization:
- In a two-input case, there are two possible paths from $x_t$ to $x_{t+1}$ (changing input 1 first, then input 2, or vice versa). The accumulation effect is decomposed along both paths.
- The final input-specific contribution is derived by taking the geometric mean of the accumulation effects across all possible permutations of input changes.
- For $m$ inputs, the method averages over all $m!$ permutations. The contribution of input $k$ is defined as the geometric mean of the projected output ratios associated with the step where input $k$ changes, averaged across all paths and both time periods ( $t$ and $t+1$ ).
Implementation: The decomposition relies on standard DEA projection mappings, $F(x, t)$ , which assign a frontier output value to any input vector. The method is compatible with various DEA orientations (e.g., output-oriented) and returns-to-scale assumptions (e.g., VRS), provided the projection mapping is well-defined. The paper addresses feasibility issues where cross-period projections (e.g., $F(x_{t+1}, t)$ ) may be infeasible if the input vector lies outside the contemporaneous technology set, recommending sample restrictions or diagnostic protocols to handle such cases.

Key Contributions

Order Invariance: The proposed decomposition yields input-specific accumulation components that are uniquely defined and invariant to the ordering of input changes. This eliminates the ambiguity inherent in sequential attribution.
Multiplicative Consistency: The product of the order-invariant, input-specific accumulation components exactly equals the aggregate accumulation term derived from the standard Malmquist index. This ensures internal coherence with the standard nonparametric productivity framework.
Theoretical Formalization: The paper formally establishes that the log of the input-specific component equals the arithmetic average of marginal log-contributions over all admissible orderings, providing a rigorous Shapley-type interpretation within a DEA frontier framework.
Reduction to Standard Cases: The method is shown to reduce to the standard Malmquist accumulation measure in one-dimensional settings or when only one input changes, ensuring methodological consistency.

Empirical Results
The methodology is applied to international productivity data from the Penn World Table (PWT 11.0) for the period 2000–2010, decomposing labor productivity growth into efficiency change, technological change, and accumulation. The study compares a one-input specification (physical capital only) against a two-input specification (physical capital and human capital).

Impact of Input Specification: Moving from a one-input to a two-input specification alters the decomposition of productivity growth. On average, the inclusion of human capital as an explicit input reduces the measured technological change (frontier shift) and increases the measured accumulation (movement along the frontier). This suggests that part of what appears as a "frontier shift" in restricted models is actually movement along an expanded frontier when human capital is explicitly modeled.
Heterogeneity: The two-input decomposition reveals distinct patterns across economies. In advanced economies (e.g., Australia, Canada, US), accumulation is driven almost entirely by physical capital ( $HACCUM \approx 1$ ). In contrast, emerging economies (e.g., Pakistan, Cambodia) show significant contributions from human capital accumulation.
Numerical Stability: The empirical checks confirm that the path-symmetrized decomposition is numerically stable and satisfies the multiplicative identity ( $ACCUM = KACCUM \times HACCUM$ ) without the instability associated with path-dependent attributions.

Significance and Claims
The paper claims to offer a transparent and robust tool for examining the internal structure of input accumulation in DEA–Malmquist analysis. By resolving the order-dependence problem, the method allows researchers to distinguish the contributions of different inputs (such as physical and human capital) without arbitrary modeling choices affecting the results. The approach is presented as a generalizable extension to the standard DEA–Malmquist framework, applicable to firm-, industry-, or regional-level analyses with multiple inputs, provided the necessary projection mappings are feasible. The author emphasizes that the method does not impose parametric functional forms but relies on the nonparametric nature of DEA, making it a flexible addition to existing productivity accounting literature.

Technical Summary: An Order-Invariant Decomposition of Input Accumulation in DEA–Malmquist Productivity Analysis

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