📈 economics

Integrating equity and productivity in health evaluation

This paper proposes a unified, tractable framework for evaluating health outcomes that jointly integrates equity and productivity by introducing a broader class of evaluation functions grounded in normative criteria like scale independence and Pigou-Dalton transfer principles.

Original authors: Kristian S. Hansen, Juan D. Moreno-Ternero, Lars P. Østerdal

Published 2026-05-05

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Kristian S. Hansen, Juan D. Moreno-Ternero, Lars P. Østerdal

Original paper licensed under CC BY 4.0 (http://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 you are a city planner trying to decide how to spend a limited budget on public parks. You have two main goals:

Health: Making sure everyone is happy, healthy, and alive.
Productivity: Making sure everyone has the energy and ability to work, contribute to the community, and keep the economy running.

For a long time, economists have used a ruler called a QALY (Quality-Adjusted Life Year) to measure the "Health" goal. It's like counting how many years people live, but giving less weight to years spent in pain or sickness. If a policy gives a sick person 5 healthy years, that's a "5."

But this ruler has a blind spot. It ignores the Productivity goal. It doesn't care if those healthy years are spent working, volunteering, or caring for a family. Conversely, some newer rulers (like PALYs) focus heavily on how much work a person can do, but they might ignore the quality of their life or fairness to the sick.

This paper introduces a new, super-ruler that tries to measure both at the same time, while also making sure the rules are fair.

The Core Idea: The "Healthy Productive Year" (HPYE)

The authors suggest we stop looking at health, work, and time as separate things. Instead, imagine every person has a "score" that combines all three. They call this a Healthy Productive Year Equivalent (HPYE).

Think of it like a video game character's "power level."

If you are fully healthy, working at 100% capacity, and have 40 years left, your power level is high.
If you are fully healthy but can only work at 50% capacity, your power level is lower.
If you are sick and can't work at all, your power level is very low.

The paper asks: How do we add up these power levels for an entire population to decide which policy is best?

The Problem with "Just Adding Them Up"

If you just add up everyone's scores (a method called "utilitarianism"), you might end up with a weird result. Imagine two groups:

Group A: One person has a massive score of 100. Everyone else has a score of 0. Total = 100.
Group B: Everyone has a score of 20. Total = 100.

A simple "add them up" rule says these groups are equal. But most people feel Group B is better because it's fairer. The person in Group A with the score of 0 is being left behind.

The paper argues that we need a rule that says: "Giving a little boost to someone with a low score is more valuable than giving a big boost to someone who is already doing great." This is the "Equity" part of their framework.

The New Rules of the Game

To build this fair, combined ruler, the authors set up a set of logical rules (axioms). Here are the most important ones, translated into everyday language:

1. The "No Name-Calling" Rule (Anonymity)
It doesn't matter who gets the health or the job. If Person A gets a boost and Person B loses one, it's the same as if Person B got the boost and Person A lost one. We only care about the distribution of the scores, not the names on the list.

2. The "Fair Share" Rule (Pigou-Dalton)
This is the heart of the equity argument. Imagine two people: one has a lot of "healthy productive time," and one has very little. If we take a tiny bit of time from the rich person and give it to the poor person (without changing who is richer), the whole society is better off. The paper insists that any good ruler must agree with this.

3. The "Unit of Time" Rule (Scale Independence)
It shouldn't matter if we measure life in years, months, or days. If we double the time unit for everyone, the ranking of who is doing better shouldn't change. This ensures the math works no matter how you count.

4. The "Zero is Zero" Rule
If someone has zero years left to live, their health or job status doesn't matter anymore. They contribute nothing to the total score.

The Result: A Family of "Smart Rulers"

By combining these rules, the authors found that there isn't just one perfect ruler. Instead, there is a family of rulers that look like this:

$\text{Total Score} = \sum (\text{Health} \times \text{Work} \times \text{Time}^\gamma)$

Health: How good is your health? (0 to 100%)
Work: How productive are you? (0 to 100%)
Time: How many years do you have?
The Magic Number ( $\gamma$ ): This is the "Fairness Knob."

How the "Fairness Knob" works:

If you turn the knob to 1, you are a pure "adder." You just sum up the total years. You don't care who has them.
If you turn the knob lower (closer to 0), you become very sensitive to inequality. You start valuing the years of the person with the shortest life much more than the person with the longest life.

The paper shows that you can create different versions of this ruler:

The Health-Only Ruler: Ignores work, focuses only on health and time.
The Work-Only Ruler: Ignores health, focuses only on productivity and time.
The Hybrid Ruler (PQALY): Mixes health and work together. This is the paper's main innovation. It says, "We care about your health and your ability to work, and we want to be fair to the people who have the least of both."

Why This Matters (According to the Paper)

The authors don't claim this is a magic wand that will fix the healthcare system tomorrow. Instead, they provide the mathematical blueprint.

They prove that it is logically possible to create a system that:

Counts both health and productivity.
Treats the "worst off" (those with poor health and low productivity) with extra care.
Follows strict logical rules so the results aren't just random guesses.

They show that by adjusting the "Fairness Knob" ( $\gamma$ ), policymakers can choose how much they care about equality versus total output. If a society wants to prioritize the sick and the unemployed, they turn the knob down. If they want to maximize total economic output, they turn it up.

The Bottom Line

This paper is like an architect drawing up the blueprints for a new kind of measuring tape. Old tapes measured only length (health). Some new tapes measured only width (productivity). This paper designs a 3D measuring tape that captures length, width, and depth, while adding a special feature that ensures the tape doesn't ignore the people who are the smallest.

It doesn't tell you which setting to pick (that's a political choice), but it gives you the tools to understand exactly what happens when you change the settings.

1. Problem Statement

Health interventions impact not only the level and distribution of health but also the productive capacity of individuals and populations. Traditional health evaluation tools, such as Quality-Adjusted Life Years (QALYs) and Disability-Adjusted Life Years (DALYs), focus primarily on health outcomes (longevity and quality of life) and often overlook broader societal concerns like equity and productivity. Conversely, Productivity-Adjusted Life Years (PALYs) and the more recent Productivity- and Quality-Adjusted Life Years (PQALYs) (Hansen et al., 2024) incorporate productivity but often rely on utilitarian aggregation, which fails to address distributive justice or equity concerns (e.g., prioritizing the worst-off).

There is a critical gap in the literature for a unified, axiomatic framework that simultaneously:

Integrates health and productivity as distinct but interacting dimensions.
Incorporates equity sensitivity (inequality aversion) regarding the distribution of both health and productive capacity.
Provides a normative foundation for policy decisions where trade-offs between efficiency (productivity) and equity are central.

2. Methodology

The authors employ an axiomatic approach rooted in welfare economics and social choice theory. They extend the model introduced by Hansen et al. (2024) by defining a population of $n$ individuals, where each individual $i$ is characterized by a profile $d_i = (a_i, p_i, t_i)$ :

$a_i \in A$ : Health state (with a benchmark of full health $a^*$ ).
$p_i \in [0, 1]$ : Productivity level (relative, where 1 is maximal).
$t_i \in \mathbb{R}_+$ : Remaining lifetime.

The core of the methodology involves defining a Healthy Productive Years Equivalent (HPYE), denoted as $f(a_i, p_i, t_i)$ . This represents the amount of time an individual would need to live in full health ( $a^*$ ) and maximal productivity (1) to be indifferent to their actual profile $(a_i, p_i, t_i)$ .

The authors then characterize Social Evaluation Functions (SEFs), $E: D \to \mathbb{R}$ , which rank distributions of these profiles. They derive specific functional forms by imposing a set of normative axioms:

Core Structural Axioms (CORE):

Anonymity: Permuting individuals does not change the ranking.
Separability: The ranking of two distributions depends only on the subgroup where profiles differ.
Continuity: Small changes in attributes lead to small changes in evaluation.
Social Zero: Individuals with zero lifetime do not influence the evaluation.
Full Health/Productivity Superiority: Improving health or productivity cannot worsen the evaluation.
Lifetime Monotonicity: At full health and productivity, longer life is strictly better.
Positive Lifetime Desirability: Moving from zero to positive lifetime is desirable.
Pigou-Dalton Transfer at Full Health and Productivity (PDTFHP): A transfer of lifetime from a richer (longer-lived) to a poorer (shorter-lived) individual, both at full health/productivity, improves the distribution (capturing equity sensitivity).

Additional Axioms for Specific Forms:

Time Scale Independence (TSI/TSIFHP): Rankings are invariant to rescaling time units (e.g., years to months), implying relative duration matters, not absolute units.
Productivity Scale Independence (PSI): Rankings are invariant to rescaling productivity levels.
Independence Axioms (PI, HI): Evaluations are independent of productivity (PI) or health (HI) under specific conditions.
Transfer Invariance Axioms (PICHT, PICT, PDTFHCT): Specific conditions regarding how productivity transfers are treated relative to health and time.

3. Key Contributions and Results

The paper derives a hierarchy of evaluation functions, moving from general equity-sensitive forms to specific power-function representations.

A. General Equity-Sensitive Framework (Theorem 1)

The authors first establish that any evaluation function satisfying the CORE axioms can be represented as a Generalized HPYE Evaluation Function:
$E_g[d] = \sum_{i=1}^n g(f(a_i, p_i, t_i))$
where $g$ is a strictly increasing and strictly concave function. The concavity of $g$ captures the equity sensitivity (prioritarianism), placing higher marginal weight on gains for those with lower HPYEs.

B. Power-Form Representations

By imposing Time Scale Independence (TSI), the authors narrow the class of functions to power forms. This yields several specific families:

Equity-Sensitive Power HPYE (Theorem 2):
$E_{\gamma f}[d] = \sum_{i=1}^n (f(a_i, p_i, t_i))^\gamma, \quad 0 < \gamma < 1$
Here, $\gamma$ is the inequality aversion parameter. Lower $\gamma$ implies stronger aversion to inequality in healthy productive years.
Equity-Sensitive Productivity- and Quality-Adjusted Power Lifetime (Theorem 3):
$E_{\gamma \phi}[d] = \sum_{i=1}^n \phi(a_i, p_i) t_i^\gamma$
This separates the time component (raised to power $\gamma$ ) from a weighting function $\phi$ that adjusts for health and productivity.
Special Cases (Propositions 1 & 2):
- Power QALY: If productivity is ignored (PI axiom), the function becomes $\sum q(a_i)t_i^\gamma$ .
- Power PALY: If health is ignored (HI axiom) and specific productivity invariance axioms hold, the function becomes $\sum p_i t_i^\gamma$ .
Integrated Compromises (Theorems 4, 5, 6):
- Power PQALY: A multiplicative integration of health, productivity, and time: $\sum q(a_i) p_i t_i^\gamma$ .
- Power QALY-PALY: A convex combination of Power QALY and Power PALY.
- Power QALY-PQALY: A hybrid combining Power QALY and Power PQALY.
Bi-Power PQALY (Theorem 7):
The most flexible form, allowing distinct inequality aversion parameters for lifetime ( $\gamma$ ) and productivity ( $\varepsilon$ ):
$E_{\varepsilon \gamma}[d] = \sum_{i=1}^n q(a_i) p_i^\varepsilon t_i^\gamma$
This allows policymakers to tune sensitivity to productivity inequality separately from lifetime inequality.

C. Empirical Illustration

The authors apply these functions to a numerical example involving two distributions ( $d_\Omega$ and $d_\Phi$ ) with five individuals. They demonstrate that:

The ranking of distributions is non-monotonic with respect to the equity parameter $\gamma$ .
Different functional forms (e.g., multiplicative vs. additive) yield different rankings for the same distributions depending on parameter values.
This highlights the necessity of explicitly choosing a normative framework rather than relying on a single "default" metric.

4. Significance and Implications

Theoretical Contribution:

Unification: The paper provides the first full axiomatic characterization of equity-sensitive aggregation where individual equivalents are defined jointly over health and productivity.
Beyond Utilitarianism: It moves beyond the utilitarian aggregation of Hansen et al. (2024) by introducing explicit distributive principles (Pigou-Dalton transfers) that rule out unweighted aggregation.
Flexibility: The framework offers a "menu" of admissible evaluation criteria, allowing for distinct normative weights on health and productivity dimensions.

Policy Relevance:

Occupational Health & Safety (OHS): The framework is particularly relevant for OHS, where interventions affect both worker well-being and economic output.
Resource Allocation: It offers a rigorous method to balance efficiency (productivity) and equity (distribution), addressing the criticism that standard QALYs ignore the "worst-off."
Decision Support: By showing how rankings change with parameters (e.g., $\gamma$ , $\varepsilon$ ), the model helps policymakers understand the trade-offs inherent in their choices.

Limitations and Future Directions:

Normative Judgments: The choice of specific functions and parameters requires societal consensus on trade-offs, which is difficult to elicit.
Data Requirements: Operationalizing these metrics requires robust data on productivity (employment, caregiving), which is often lacking.
Static Model: The current model is static; future work could extend it to dynamic trajectories over the life course.
Health States Worse than Death: The model assumes positive lifetime desirability, which complicates the inclusion of health states worse than death (a common feature in HRQoL literature).

In conclusion, this paper establishes a rigorous, flexible, and normatively grounded framework for evaluating health interventions that explicitly accounts for the dual goals of improving health equity and sustaining economic productivity.