Deep Linking Financial and Energy Accounting

We develop a conceptual framework for integrated accounting that imposes on certain non-financial disclosures the same double-entry balance constraints that apply to conventional financial statements. We identify the key ingredients required for a rigorous multidimensional accounting framework in terms of concepts, postulates and design choices, and we illustrate these ideas with a worked-out example of linking financial and energy accounts.

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Integrated Energy Accounting is keeping track and reporting on an entity’s detailed energy footprint (primary inputs, transformations and waste generation) not as an addendum to financial accounting and reporting but as a deeply-linked extension that is subject to the same level of rigor.

The central design is the use of multidimensional double-entry bookkeeping which tracks additional quantitative information characterizing economic objects beyond their monetary values. This choice ensures the enforcement of both classic balance sheet constraints and the applicable energy conservation laws.

This blog post offers an overview of the overall design and principles involved, the full exposition is available in the white paper.


Accounting, along with a number of related sub-disciplines across finance, economics and risk management forms a pillar of the quantification of economic life. It is an essential information technology construct, and it is difficult to imagine how complex modern society could function without some version of accounting.

The collection and reporting of significant new facets of economic activity (collectively termed non-financial data or sustainability accounting) is already in full swing. An important emerging issue that is already much discussed is the ability of achieving management objectives on the basis of multiple and disconnected data points (e.g. financial versus environmental or social “returns”).

New frictions such as greenwashing are already a concern. Making sure that new reported accounting dimensions are at least as reliably produced as existing ones and that such new facets are offering a consistent and usefully integrated view of economic activities is a strong motivation. Towards that end, sketching a deeper link between the old financial accounting schemata and the newly relevant sustainability dimensions is the objective for this white paper.

Focusing specifically on integrated financial-energy accounts (as distinct from GHG emissions or more general environmental impact) has three-fold motivation:

  • The energy footprint of economic activity is very important in a sustainability context as it drives most emissions
  • Energy usage information can be readily quantified without losing much of its semantic content, and
  • Energy is not an economic externality (as are for example GHG emissions) but a significant factor of production. More informed accounting of energy flows through the economy is thus a competitive advantage.

Core Ideas

Central to the generalization we explore here is the notion of accounting for diverse measurement qualities: quantifying the same economic phenomenon using different lenses (financial, energetic etc).

Each dimension being accounted for introduces a new quality measure of an underlying economic artifact. In conventional accounting all measured values populating an entity’s accounts are scalars (that is, single values) expressed in financial (monetary / currency) units. Extending the accounting system introduces additional qualities or dimensions and the corresponding measurement methodologies required to establish reliably such values.

Simplistically the idea involves stacking alternate measurement views of the same reality, somewhat akin to a color photograph that is optically produced by overlaying three primary colors (red, green and blue). It is the different proportions of primary colors that give rise to the final visual perception. In analogy, it is the complete set of values that characterises fully, e.g. an inventory account, a sales account or a profit and loss account. Beyond monetary considerations, the amounts of energy that are associated with an account either directly or indirectly (as embodied energy) link that economic activity with tangible real world impacts.

The extended accounts of an entity’s general ledger require that additional account slots are populated with numerical values not expressed in currency units (e.g.Euros) but other units (e.g., kilos, liters) which in the case of energy would be Joules.

The energy qualities that will be measured and reported play a central role in the formulation of integrated energy accounting. The minimal design must on the one hand capture real energy flows, thereby linking with actual environmental states and impacts and on the other hand incorporate energy provenance as this is essential for fair attribution of benefits and responsibilities. We will use the term physical energy to denote the first measurable while embodied energy is the second measurable.

Multi-Dimensional Accounts

Mathematically an integrated accounting space is a multidimensional version of the standard approach:

$$ \begin{equation} V = \begin{bmatrix} V_1 & V_2 & \ldots & V_N \\
\hline v_{11} & v_{12} & \ldots & v_{1N} \\
\ldots & \ldots & \ldots & \ldots \\
v_{q1} & v_{q2} & \ldots & v_{qN} \end{bmatrix} \end{equation} $$

where the first row is just the label of the account and each account has a column of measured qualities $v^{a}_{q}$.

For concreteness we might think of a stylized entity with the following list of accounts:

Account ID Account Name Account Description
V1 C Cash
V2 F Plants: Solar Panel Array and Widget Factory
V3 S Energy Stock
V4 M Raw Materials
V5 I Widget Inventory
V6 P Accounts Payable
V7 L Bank Loan
V8 K Equity

where in principle all accounts have three associated measurements (expressed in their respective units):

$$ \begin{equation} V = \begin{bmatrix} C & F & S & M & I & \bf{P} & \bf{L} & \bf{K} \\
120 & 1000 & 300 & 150 & 600 & 0 & 0 & 2170 \\
0 & 0 & 500 & 0 & 0 & 0 & 0 & 500 \\
0 & 100 & 0 & 600 & 700 & 0 & 0 & 1400 \end{bmatrix} \end{equation} $$

Multi-Dimensional Balance Equations

The quintessence of rigorous accounting is double-entry bookkeeping (DEB) which ensures that accounts are always balanced. Mathematically this means that the sum of all accounts is preserved to be zero at all times.

$$ \begin{equation} \sum_{a}^{N} V^{a}_{q, t} = 0 , \text{for all } q \end{equation} $$

While in standard accounting balance equations only involve a single dimension $q=1$ (the financial leg), more general systems require as many balance equations as the range of $q$. If we write out explicitly the legs of the balance equation for our system where we track financial value, physical energy and embodied energy we have:

$$ \begin{align} \sum_{a}^{N} M^{a}_{t} & = 0 ,& \text{Total Financial Value Balance} \\
\sum_{a}^{N} \mathcal{E}^{a}_{t} & = 0 ,& \text{Total Physical Energy Balance} \\
\sum_{a}^{N} E^{a}_{t} & = 0 ,& \text{Total Embodied Energy Balance} \end{align} $$

The system of accounts is updated via transactions $T$ that must satisfy the same balance equations. Thus, when added to existing accounts, the new accounting state keeps satisfying the balance equations:

$$ \begin{align} V^{a}_{t+1} = V^{a}_{t} + T^{a}_{t} ,& \text{ Posting a transaction at time t} \end{align} $$

Taken together, the balance conditions and the update equation circumscribe the essential mechanics of the multidimensional accounting system. The integrated (financial and energy) state of an entity is updated by applying atomic transactions that modify all affected measured or attributed qualities of an account preserving balances across all measured dimensions.

Energy Conservation

Energy conservation in our context is a statement about stylized energy flows across the entity’s economically defined perimeter as expressed through the asset and liability decomposition. It is captured by the constraints on transaction values affecting physical and embodied energy:

$$ \begin{align} \sum_{a}^{N} \delta \mathcal{E}^{a}_{t} & = 0 ,& \text{ Physical Energy Flow Balance} \\
\sum_{a}^{N} \delta E^{a}_{t} & = 0 ,& \text{ Embodied Energy Flow Bala nce} \\
\sum_{a \in A} \delta \mathcal{E}^{a}_{t} + \sum_{a \in A} \delta E^{a}_{t} & = 0, & \text{ Physical to Embodied Energy Transformations} \end{align} $$

Integrated Accounting Principles

We can summarize a number of important principles or design choices that underpin the integrated energy accounts proposal as follows:

Principle 1: Multi-value Accounts and Transactions following DEB. Accounts and Transactions encoding mutations of accounts are generalized to hold additional measurement values for each economic artifact recognized as financially or energetically significant. The integrated (financial and energy) state of an entity is updated by applying atomic transactions that modify all affected measured or attributed qualities of an account. Accounts and Transactions satisfy double-entry balance constraints for each measured dimension.

Principle 2: Alignment of Scope. Financial accounts define the basic perimeter and scope of the integrated accounting system, in particular the asset and liability segmentation is adopted for the attribution of energy values to liabilities. Additional accounts are introduced to ensure the integrity of the extended system as required

Principle 3: Holistic and Consistent Measurement. The measurement process defines additional methodologies for quantifying a consistent set of energy qualities (minimally physical and embodied energy) for each one of the recognized accounts.

Principle 4: Fair Attribution. Attribution methodologies enforce consistent overall asset and liability balance also when the causal links of assets with liabilities are not obvious.

Principle 5: Enforcement of Conservation Laws. Energy conservation laws are expressed by requiring that energy qualities in transactions balance when summed across all relevant energy qualities (stoichiometric balance).

Principle 6: Role of Equity. The equity relation retains its role as residual and perpetual liability. In the absence of other liabilities all changes to an entity’s stock of energy are attributed to this account.

Principle 7: Integrated Reporting. Periodic reports identify accounts of specific interest and explain their start and end-of-period changes. This “explain” function is constructed by aggregating transactions over classes of similar nature. Reports present the monetary and energy facets of the entity as reconcilable parallel streams.

Comparing and Contrasting Monetary and Energy Dimensions

The following table aims to provide a summary of similarities, analogies and differences between the financial and energy dimensions of accounts.

Concept Financial Dimension Energy Dimension
Accounting Entity An economic agent that uses accounting to track and report a financial view of its economic state An economic agent that uses accounting to track and report a physical and embodied energy view of its economic state
Measurement Determination of verifiable financial value from a menu of agreed methodologies Determination of verifiable physical and embodied energy values from a menu of agreed methodologies
Value Sign Positive when owned, negative when owed Positive when owned, negative when owed
Economic Production Uses input economic resources to create added financial value. There is no concept of financial waste but there is a concept of value destruction Uses input physical energy and materials to create added energy value (lower entropy artifacts). Produces waste (higher entropy artifacts) which is captured as part of embodied energy
Assets Stock accounts of elements that are expected to inject economic value into the entity in the future (they indicate opportunity for value creation) Stock accounts of elements that are expected to inject energy potential into the entity’s inventory (they indicate opportunity for useful work)
Liabilities Stock accounts of elements that will extract economic value from an entity in the future. They indicate responsibility to external parties to meet liabilities Stock accounts of elements that get attributed the energy potential of an entity’s inventory. They indicate the responsibility of external parties for an entity’s energy profile
Equity as Special Contract Liability (Perpetual, Residual Interest) Is attributed any residual financial value. Concept of Net Worth Is attributed any residual physical and embodied energy. Concept of Net Energy Liability
Statement of Position Articulates financial balance across an entity’s asset/liability boundary Articulates energy balances across an entity’s asset/liability boundary
Statement of Performance Explains the change in net worth. Temporary value accounts group transactions by their nature to facilitate explaining Explains the change in net energy liabilities. Temporary energy accounts group transactions by nature
Double Entry Bookkeeping (DEB) All transactions and accounts must balance financial value All transactions and accounts must balance physical and embodied energy value
Conservation Laws Not applicable Inflows, outflows, storage and conversion of physical energy to embodied energy must balance across energy qualities
Measurement Identities Cash has measured value equal to face value Physical energy is converted to an equal amount of embodied energy
Medium of Exchange Cash (Money) Barter (in principle)
Medium of Exchange Accounting Cash Flow Accounting and Statement Not applicable
Taxation Tax Liabilities on Financial Flows Tax Liabilities on Physical Energy or Embodied Energy (Putative)
Special Accounting Entities Banking Sector (Money creation) Energy Sector (Primary energy creation)

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