Value per Joule: Foundations of an Energy-Normalized Economics

Energy productivity and useful work

The reciprocal pair “energy intensity” and “energy productivity” has long been used at sector and economy scale. If $Y$ is a monetary output and $E$ is energy, then $Y/E$ is commonly called energy productivity, while $E/Y$ is energy intensity. The label does not settle whether $Y$ is gross output, value added, welfare, or a volume index, nor whether $E$ is primary or final energy. Patterson documents exactly this diversity [1]. The International Energy Agency likewise treats indicator construction as a matter of decomposition, activity, structure, and end-use efficiency rather than one universal number [5]. Ang’s practical guide to logarithmic mean Divisia decomposition shows how an aggregate change can be separated into activity, structure, and intensity terms [22].

Useful-work economics moves the physical account closer to end-use services. Ayres and Warr model production using capital, labor, and useful work [2]. Warr et al. construct long historical series by mapping energy carriers through conversion devices to useful work [3]. Warr and Ayres extend the account to information and communication technologies [4]. These contributions matter for the denominator and for the physical stages that precede economic outcomes. They do not imply that useful work, information, and welfare have a common natural unit. Their thermodynamic framing sits in a longer ecological-economics tradition that includes Georgescu-Roegen’s account of entropy and economic process [23].

The present framework is compatible with that literature. Exergy, final energy, and useful work can appear as separately typed nodes in an accounting chain. The framework’s proposed contribution is to join physical boundaries, counterfactual value, attribution, and uncertainty in one comparison contract. It does not rename existing energy productivity as a new theory.

Lifecycle assessment and environmental-economic accounts

ISO 14040 places goal and scope definition before lifecycle inventory and interpretation [6]. A functional unit provides the reference for comparison, while a system boundary determines which unit processes belong to the product system. ISO 14044 adds requirements and guidance for lifecycle inventory and critical review [7]. Rebitzer and coauthors survey the corresponding framework, goal-and-scope, and inventory practice [21]. Those principles carry directly to energy denominators. An operational wall-energy figure and a lifecycle energy figure are not rival estimates of one hidden number. They answer different scoped questions.

The System of Environmental-Economic Accounting Central Framework integrates physical flows, environmental assets, and economic accounts [8]. SEEA-Energy organizes energy resources and products in physical supply and use tables [9]. Its economic-account side is consistent with the production and accounting structure of the System of National Accounts [25]. Transformation inputs and outputs can both appear in a gross table, so analysts need bridge tables or net presentations when forming an intensity denominator. Adding primary fuel to the electricity made from that fuel would count one flow twice.

Our tuple borrows the discipline of goal, scope, boundary, and allocation from lifecycle assessment, then adds an explicit counterfactual and value functional. It also keeps primary, final, operational, and lifecycle energy conventions distinct. Paper 9 of the larger research program develops the national-account implications; this paper confines itself to the measurement foundation.

Welfare, index numbers, and composite indicators

Gross domestic product is a production measure, not a complete welfare functional. Fleurbaey reviews the long search for welfare measures beyond GDP and finds multiple defensible approaches rather than one settled substitute [10]. Sen’s treatment of collective choice supplies part of the social-choice background to that measurement problem [24]. Fleurbaey and Blanchet give a book-length treatment of welfare and sustainability measurement [11]. Stiglitz, Fitoussi, and Durand similarly argue for dashboards and distributional information alongside aggregate production [12].

Index-number theory shows that aggregation formulas embody assumptions about the underlying aggregator. Diewert’s exact and superlative indices do not evade this point; they make the connection between a formula and a flexible aggregator explicit [13]. The OECD and Joint Research Centre handbook treats normalization, weighting, aggregation, uncertainty, and sensitivity analysis as separate design steps for composite indicators [14]. Rankings can change when weights or aggregation rules change.

Value per joule inherits these concerns. Dividing a composite value index by energy does not make the weights objective. The denominator can sharpen the resource question, but it cannot choose a social welfare functional.

Counterfactuals and attribution

An incremental-value numerator requires a baseline. The potential-outcomes tradition distinguishes causal effects from observed associations [15, 16]. A unit cannot be observed both under an intervention and under its alternative, so identification rests on design and assumptions. Imbens and Rubin give a systematic treatment of randomized and observational designs [17]. Manski shows how partial identification can report informative bounds when point identification is not defensible [18].

The tuple therefore includes both a counterfactual $c$ and an attribution rule $a$. The counterfactual states what would happen otherwise. The attribution rule states how evidence connects observed data to that contrast, and how shared outputs or burdens are assigned. Keeping these fields separate prevents a frequent error: writing down a causal-looking numerator while estimating only an observational ratio.

Why seven fields

Each tuple component blocks a distinct ambiguity. Boundary $b$ says what equipment, infrastructure, upstream process, and downstream process belong to the study. Counterfactual $c$ determines incrementality. Functional $W$ defines the numerator. Horizon $\tau$ controls when benefits and energy enter. Convention $e$ distinguishes stages of energy accounting. Attribution $a$ allocates shared causes, assets, and burdens. Uncertainty model $\pi$ states what is random, how dependence is represented, and what interval accompanies the point estimate.

None can be inferred reliably from the others. A lifecycle boundary does not select a counterfactual. A randomized counterfactual does not decide how to allocate a shared data center. A price-year label does not disclose a welfare functional. The tuple is intentionally redundant with a good study protocol: its purpose is to make omissions visible.

A practical gate

Before ranking two systems, an analyst should answer the following questions.

1. Do the functional units describe the same service?

2. Do boundaries include compatible physical components?

3. Are energy stages identical or connected by a documented bridge?

4. Do counterfactuals describe the same alternative?

5. Does $W$ use the same population, prices, weights, and discount rule?

6. Do horizons match?

7. Are idle energy, failures, shared infrastructure, and embodied burdens allocated under compatible rules?

8. Are uncertainty intervals based on compatible estimands?

If one answer is no, the comparison is not automatically useless. It becomes a sensitivity exercise or an OBSTRUCTED direct comparison. The analyst can report each result in its own class, derive a bridge, or show a range over plausible tuples. What is not justified is a point ranking that suppresses the difference.

Valuation weights

Suppose a system and its counterfactual produce $m$ heterogeneous outcomes. Write the componentwise increment as $\Delta x(s\mid c)=x(s)-x(c)$. For nonnegative weights $w\in\mathbb{R}_{\geq 0}^m$, consider the linear functional $W_w(x)=w^\top x$. Define the incremental-outcome-per-energy vector $$z(s)=\frac{\Delta x(s\mid c)}{E(s)},$$ where division acts componentwise.

PROVED. The theorem does not say every small change of weights reverses a ranking. It says a universal scalar order is unavailable for non-dominating vectors unless the admissible valuation class is restricted. The restriction may be ethically or institutionally justified, but it must be published.

Dominance is the strongest defensible cross-weight conclusion. When it fails, a Pareto frontier communicates more than a single league table.

System boundaries

PROVED, conditional on the accounting admissibility of the boundary expansion. The theorem is not permission to add arbitrary energy to a disfavored system. The added burden must belong to the declared expanded boundary under a consistent allocation rule. Examples include manufacturing energy for dedicated hardware or a facility load omitted from a device measurement.

Counterfactuals

PROVED as a non-identification construction. Empirical counterfactuals cannot be chosen after seeing the desired answer. The construction shows why observed revenue per joule alone does not identify incremental value per joule. Design, institutional knowledge, or defensible bounds must restrict the counterfactual class.

Time horizons

PROVED. Delayed maintenance costs, learning effects, degradation, and asset life can generate the same structure. A horizon should therefore be chosen from the decision problem, not from whichever endpoint favors a technology.

Shared burdens and attribution

PROVED. An admissible allocation still requires a rationale, such as measured marginal use, capacity reservation, run time, or causal responsibility. Equal splitting, revenue shares, and peak-capacity shares are not interchangeable merely because they sum to the same total.

No universal scalar order

PROVED under the stated valuation class. The theorem is deliberately weaker than a social-choice impossibility theorem. It needs no claim about collective preference aggregation. It only records a fact about scalarizing heterogeneous outcomes.

Value weights

Consider two stylized systems that each use 100 joules. Relative to a zero-vector counterfactual, system A adds nine verified task completions and two units of avoided loss. System B adds three verified task completions and eight units of avoided loss. Their incremental-outcome-per-energy vectors are $$z(A)=(0.09,0.02),\qquad z(B)=(0.03,0.08).$$ If the functional assigns one index point to each completed task and zero to avoided loss, A ranks first. If it assigns one point to each avoided-loss unit and zero to task count, B ranks first. Neither vector dominates.

The correct response is not to average the dimensions by default. One can publish both components, state a reason for weights, and show a sensitivity region. For linear weights $w_T,w_L\geq0$, A ranks first when $$0.09w_T+0.02w_L>0.03w_T+0.08w_L,$$ or $w_T>w_L$. B ranks first when $w_T<w_L$, and they tie when the weights are equal. The threshold is transparent.

Operational and lifecycle boundaries

Now give both systems the same realized value of ten currency units. Under an operational boundary, A uses 40 joules and B uses 50. Their ratios are 0.25 and 0.20 currency units per joule, so A ranks first. An expanded boundary assigns 100 joules of embodied energy to A and 20 to B. The lifecycle denominators become 140 and 70. The ratios become approximately 0.0714 and 0.1429, so B ranks first.

This example is COMPUTATIONAL. It verifies the construction in 16; it is not a lifecycle assessment of a real technology. An empirical study would need inventories, lifetimes, utilization, allocation, geography, and uncertainty.

Physical efficiency versus realized value

Suppose heat system H converts 90 percent of final electricity into useful heat, while system L converts 70 percent. On a matched physical functional unit and boundary, H is more physically efficient. Now suppose H serves a process that often produces inventory with no buyer, while L serves a temperature-sensitive repair whose completion avoids a costly outage. L can have higher expected realized value per joule despite lower heat-delivery efficiency.

There is no paradox. Physical efficiency answers how much useful heat leaves the device for a given physical input. Economic VPJ also depends on the use, counterfactual, prices or welfare weights, reliability, and attribution. The physical result remains true within its domain. It is one input to the economic account, not a complete valuation.

The danger of a gross ratio

Two sites report annual revenue and electricity: $$\frac{Y_A}{E_A}=12\ \text{currency/MJ},\qquad \frac{Y_B}{E_B}=9\ \text{currency/MJ}.$$ The first site appears better. But suppose the intervention studied at A replaced a process that already earned 11 currency/MJ, while B replaced one that earned 3 currency/MJ, on matched energy bases. The incremental comparisons are one and six currency/MJ. Gross output and incremental contribution answer different questions.

The numerical subtraction alone does not establish causality. The baselines would need evidence. A randomized rollout, credible natural experiment, engineering model, or partial-identification bound could support them. The example’s role is to make the estimand distinction concrete.

Three different ratios

Let $V$ and $E>0$ be random. Analysts may encounter: $$\theta_1=\frac{\mathbb{E}[V]}{\mathbb{E}[E]},\qquad \theta_2=\mathbb{E}\left[\frac{V}{E}\right],\qquad \theta_3=\frac{\sum_i V_i}{\sum_i E_i}.$$ The third is a sample ratio estimator for the first under common sampling conditions. The second is the mean of unit-level ratios. In general, $\theta_1\neq\theta_2$. If high-value observations also use more energy, an unweighted mean of unit ratios can answer a particularly different question from an aggregate ratio. In the notation of 7, $V$ is $\Delta W(s\mid c)$ and $E$ is $E_b^e(s;\tau)$; this subsection relaxes the definition’s default treatment of the denominator as fixed.

PROVED. The uncertainty component $\pi$ must name the estimand before it names a confidence-interval algorithm.

Sources of uncertainty

A useful uncertainty ledger separates:

1. measurement error in meters, prices, and outcome records;

2. sampling variation across tasks, days, sites, or users;

3. model uncertainty in counterfactual and causal assumptions;

4. scenario uncertainty in future demand, lifetime, and discounting;

5. normative uncertainty over weights and social standing;

6. boundary uncertainty over omitted or shared processes.

The first two may yield to repeated measurement and conventional intervals. The last four often require sensitivity analysis, scenario sets, or bounds. A narrow bootstrap around one selected model does not represent uncertainty about the model or value functional. Saltelli and coauthors provide a general framework for global sensitivity analysis when uncertain inputs interact [19].

Recommended reporting

For an empirical ratio, report the numerator and denominator separately before the quotient. Include their covariance when both are estimated from the same sample. If the denominator is noisy or can approach zero, ordinary symmetric intervals can fail. The present framework requires positive energy and recommends Fieller-type or bootstrap methods only after checking their assumptions [20]. When causal identification is weak, report bounds on $\Delta W$ and propagate those bounds through the positive denominator.

Sensitivity to tuple choices is not summarized adequately by one standard error. A robust report contains:

• a statistical interval conditional on the chosen tuple;

• a scenario range over defensible boundaries and horizons;

• a weight region or Pareto statement for heterogeneous outcomes;

• an identification statement for the counterfactual;

• a list of unresolved exclusions.

Step 1: state the decision and functional unit

Start with the decision the metric will inform. “Which server is more efficient?” is incomplete. A functional unit might be one thousand requests meeting a named accuracy and latency threshold, one tonne-kilometre delivered, or one year of a settlement service under a named threat model. The unit should describe a comparable service, not an easy intermediate count.

Record the population and deployment context. A benchmark mixture, geographic site, and service-level constraint can all change outcomes and energy. If the functional unit cannot be matched, stop before forming a comparative ratio.

Step 2: draw the physical boundary

List included equipment and processes. Mark the meter location. Record idle energy, warm-up, failures, retries, power conversion, cooling, networking, and shared infrastructure as included, excluded, or separately allocated. For a lifecycle study, add manufacturing, construction, maintenance, and end-of-life processes without overlapping the operational inventory.

Choose one energy convention. Device operational energy is appropriate for a low-level engineering question. Wall energy fits many deployed computing comparisons. Facility operational energy adds site overhead. Lifecycle energy adds non-overlapping embodied burdens. Primary and final energy require a bridge through conversion losses. Marginal-grid energy is indexed by location and time. These conventions should appear as enum-like labels in data, not as prose buried in a methods appendix.

Step 3: predeclare the counterfactual

Name what would happen without the system. “No energy use” is rarely a credible baseline when another process would supply the service. Specify whether the alternative is no service, manual work, an incumbent technology, delayed action, or a portfolio response.

Then state the identification strategy. Random assignment can identify an average effect for the assigned population under its assumptions. Observational designs need exchangeability, timing, exclusion, or structural assumptions. Engineering counterfactuals need validation. If none is credible, an OPEN or partially identified numerator is more honest than a precise gross ratio.

Step 4: define the value functional

Write $W$ as an auditable calculation. For a monetary functional, state:

• currency, price year, and conversion method;

• whether the measure is revenue, profit, surplus, or cost saving;

• whose costs and benefits count;

• treatment of taxes, transfers, externalities, and distribution;

• discount rate and terminal value;

• treatment of risk and catastrophic loss.

For a nonmonetary index, state component units, normalization, weights, and aggregation rule. Publish the component vector. A composite should be a view of the data, not its only surviving form.

Step 5: set horizon and attribution

The horizon should include relevant benefits, operating energy, degradation, maintenance, and replacement. If horizons differ across systems, annualize or discount with a documented bridge. Do not count a full asset benefit against one day of energy or a full manufacturing burden against one unusually small batch.

For joint production, identify dedicated and shared flows. Prefer causal or measured marginal allocation when available. Otherwise show results under more than one defensible rule. Capacity share, run time, revenue share, mass, and equal allocation answer different questions. The allocation rule belongs in the tuple because 20 shows it can decide the ranking.

Step 6: specify uncertainty before estimation

List random variables, dependence, sampling unit, missingness, and estimator. Predeclare whether the target is a ratio of expectations or an expectation of ratios. Record meter accuracy and calibration, data exclusions, and failed observations. Run sensitivity over boundary, baseline, weights, horizon, and allocation, as well as sampling noise.

Step 7: apply the comparability gate

Compare tuples field by field. If a bridge is used, publish the equation, source data, and uncertainty. Report a scalar ranking only inside a comparison class. Outside it, report a vector, Pareto frontier, scenario range, or OBSTRUCTED comparison.

What a high ratio can mean

Inside a matched tuple, a higher ratio means more expected declared incremental value per included joule. That sentence is intentionally repetitive. Remove “declared,” “incremental,” or “included,” and the claim grows beyond the measurement.

A high ratio can result from a larger causal benefit, a smaller energy inventory, or both. The decomposition matters for policy. A low-energy, low-value activity and a high-energy, high-value activity can have the same ratio but different scale, risk, and constraints. Ratios should therefore be reported beside absolute energy and absolute value.

Average and marginal questions

An average lifecycle VPJ is not the marginal value of the next joule. The latter depends on time, location, constraints, and displacement. Average metrics may support benchmarking or historical accounting. Marginal allocation requires a response function and an opportunity cost. Paper 6 of this program develops that optimization problem.

The distinction also blocks a policy error. A sector with high average value per joule need not be the best recipient of one additional joule if it is at capacity, has diminishing returns, or cannot use energy at the available node and time. Conversely, a low average ratio does not prove that every marginal use is low value.

Private and social functionals

A firm’s profit functional can be useful for operating decisions. A social welfare functional may add consumer surplus, environmental damages, public risk, and distribution. Neither should be mislabeled as the other. Taxes and prices may internalize some external effects but not all.

Reporting both functionals can reveal the source of disagreement. If a system ranks well under private profit and poorly under a social functional, the conflict is substantive. Hiding it inside one “value” number removes information that policy needs.

Empirical scope left open

This foundation paper does not estimate a national energy-productivity series, an AI benchmark, or proof-of-work assurance. Those applications need domain specific functional units and data. Their empirical status remains OPEN here. The absence is deliberate: a synthetic example should not be mistaken for a measured sector claim.

Weight regions for two outcomes

Let the normalized difference between two systems be $d=z(r)-z(s)=(d_1,d_2)$, with $d_1>0>d_2$. Normalize nonnegative weights so $w_1+w_2=1$. System $r$ ranks first when $$w_1d_1+(1-w_1)d_2>0.$$ Solving gives $$w_1>\frac{-d_2}{d_1-d_2}.$$ The right side lies strictly between zero and one. Thus the admissible simplex contains a nonempty region favoring each system and one tie point. This gives a complete sensitivity diagram for two components.

With $m>2$, the robust-order region is the intersection of the weight simplex with a half-space: $$\mathcal{W}_{r\succeq s} = \{w\in\mathbb{R}_{\geq0}^m:\mathbf{1}^{\top}w=1,\, w^\top[z(r)-z(s)]\geq0\}.$$ Publishing this polytope is often more informative than publishing one selected weight vector.

Strictly positive weights

14 uses coordinate vectors, which permit zero weights. If the admissible class requires every component to have positive weight, the reversal still holds for sufficiently small perturbations. Let $e_i$ select a coordinate where $r$ is better. Define $$w_\epsilon=(1-(m-1)\epsilon)e_i+ \epsilon\sum_{k\neq i}e_k$$ for $0<\epsilon<1/m$. Continuity of the inner product preserves the strict inequality for all sufficiently small $\epsilon$. Apply the same construction at a coordinate where $s$ is better.

Boundary threshold with unequal values

For positive $V_r,V_s,E_r,E_s$, a burden pair $K_r,K_s\geq0$ favors $s$ when $$V_r(E_s+K_s)<V_s(E_r+K_r).$$ Solving for $K_r$ gives $$K_r> \frac{V_r}{V_s}(E_s+K_s)-E_r.$$ The expression is a sensitivity threshold, not an empirical estimate. A study can place distributions or inventory intervals on $K_r$ and $K_s$, then report the probability or scenario share on either side.

Discounted horizon reversal

In 19, let the time-two discount factor be $\delta\in(0,1]$. System B’s long-horizon value is $6+8\delta$. It exceeds A’s value of 10 whenever $\delta>1/2$. Thus the reversal persists for discount rates whose one-period factor exceeds one half. The result does not select a discount rate. It identifies the threshold at which the decision changes.

Ratio decomposition

For two tuples that differ in both numerator and denominator, a logarithmic decomposition can be useful when values remain positive: $$\log\frac{\operatorname{VPJ}_2}{\operatorname{VPJ}_1} = \log\frac{V_2}{V_1} - \log\frac{E_2}{E_1}.$$ This identity separates value change from energy change. It does not attribute causality, and it fails when incremental value is zero or negative. Those cases should be reported directly rather than repaired by an arbitrary offset.