Complexity Economics and the Accounting Framework

Please cite the paper as:
Frederico Botafogo, (2017), Complexity Economics and the Accounting Framework, World Economics Association (WEA) Conferences, No. 2 2017, Economic Philosophy, Complexities in Economics


This working paper introduces a formal language to frame the concept of complexity within economic theory. The purpose is to provide a consistent analytical framework within which the varied aspects of complexity may be given expression.

The intuition underlying the framework is provided by double-entry bookkeeping and the accounting equation whereby assets equal liabilities for all possible agents in the economic system. The discussion is focused on foundational issues; markets, preferences, convexity in production, general equilibrium are not deemed to be required in describing an economic system. The basis for developing the framework is given solely by accounting theory. Linear algebra, in particular vector and tensor analyses, provide the mathematical elements required to construct said framework.

Complexity in economics is understood to imply subjective beliefs by interacting agents, who chose strategies given their particular goals, and further who update said strategies as time goes by. The resulting framework is, accordingly, dynamic and inherently uncertain. Therein, the accounting equation is seen as the expression of a solution concept, as this is defined in Game Theory. And economic value is a measurement scale which informs how distinct resource subsets relate to each other as input to output given a strategic economic process consisting of production, trade, or both.

The epistemological background which supports the framework relies on the representational theories of measurement.

The paper is organised as a list of talking points. Three appendices cover the methodological approach being used, the mathematics knowledge being required, and a short review of what the representational approach to measurement entails.

Keywords: , , ,

Recent comments


12 comment

  • Max says:

    As we have discussed previously I am very delighted to see a reasonable and comprehensive framework that does not require any G.E assumptions to function. This is much more in line with what I imagine “complex” economics to be like.

    I would like to explore this framework via agent modelling; that is, to imbue some agents with the mental model described here “goal -> prices” and to build in some simple decision making AI. When we meet up next time, let’s discuss how we can make this happen? I have in mind a simple “toyworld” scenario where we make a n x m tile terrain that agents must traverse in order to reach their predetermined goals, and in each tile they have the ability to perform simple functions like trade, pay tolls, etc. In this scenario we might be able to see how prices can be determined from goals.

  • Frederico Botafogo says:

    Thanks Max. Please notice that to deal with several agents we must first develop the notation beyond what is presented in the appendix to this working paper. The appendix refers to a single agent only. To allow multiple agents one has to partition the given partition. The process of partitioning is accounted for by introducing additional indices, as sub-indices of the given initial indices. For example, if ‘i’ index the goods at t = 1, then ‘alpha’ could index agents such that x(i,alpha) informs how much of the ith good is controlled by the alpha th agent. Note further that the DEB consistency condition now reads as a tensor contraction of both indices such that 1 is the end result of the contraction.
    I believe the notation issue is the key to a clear understanding of this framework.

  • Valerian Popkov says:

    The comprehensive whole, as well as any other one, is made up of processes, adjoining each other, of chains of such processes. Process chains form chain complexes, which are systematically analyzed by topology. A special attention topology pays to closed chains, or cycles. The cycles, as opposed to opened chains, are able to feed themselves, and therefore may exist relatively autonomously and stably. Thus, if there were no circulation of raw materials, power and money resources in gross production, no production would exist, – it would become a single-shot phenomenon.
    The cyclic structures can be met everywhere. A specialist in synergy would say, that the cycle realizes “intensification mode”. The cycle permanently assists at all manifold self-supporting phenomena. We can allege, that the composition of qualitative specificities at every structural level of wholeness precisely corresponds the set of cycles, circulating at this level. Cycles proper do form an intermediate structural level of the wholeness, allowing some individual process to function within the frame of the given level without falling down to the lower levels and without permanent appeal to more complicated processes.
    But this is not all for the ideological content of topological approach to the system. This mathematical discipline, especially algebraic topology (a hundred years ago such construction arose in the works by Henry Pointcare), distinguishes not only structures, but most important parameters of processes, taking place within those structures. Here is meant a systematic distinguishing of contravariant and covariant substances, simply speaking, flows and potentials.

    • Frederico Botafogo says:

      Reply to Prof. Popkov:
      I am unsure what the point is, which you are trying to make. You make two comments, one I cannot make sense of, the other which I agree and will try to address in due course.
      To my understanding, topology requires dense sets; my paper relies on rational numbers only and on finite vector spaces such that I cannot see how topology could be incorporated or be used to extend the present formal framework.
      I agree that cycles are key to a proper formalisation of economics. Somewhat consistent to what is discussed in another paper being presented in this conference, when cycles are properly formalised, we are given the means to frame the concept of money endogenously. I wish to address this topic in future research; I intend to introduce the imaginary dimension, i = sqrt(-1), to be able to express a wave of probability when measuring economic value.
      Thank you for sharing your thoughts.

  • Yoshinori Shiozawa says:

    I am quite disappointed by reading this paper. Is this grandiose formulation necessary to argue, for example, the production, or input-output relations?

    Productions can be expressed by using linear algebra, or better, using vectors and matrices. If A and B are respectively material input and output coefficient matrices and s is a production scale row vector, the input for the economy as a whole is expressed by s A and the output is s B and the net product is s (BA). From these relations we can discuss many things: chains of growth path, value relations, limit of growth rate, and others. See for example, my paper The Revival of Classical Theory of Values

    Much more complicated relations such as international values can be treated similarly only using above simple notions and expressions. See for example, my paper on The New Theory of International Values: An Overview

    There is no needs to use tensor calculations that Botafogo requires. (Of course, a matrix is a 1-1 mixed tensor but there is no need to introduce such concepts.) In the appendix, Botafogo consumes five pages in giving a rudimentary introduction to linear algebra, but is it necessary to do so? I believe almost all economists have once learned elements of linear algebra except the bra-ket expression introduced by Dirac and used commonly in quantum physics but rarely in mathematics and economics. Why is it necessary to introduce such “strange” expressions for economists? Ordinary matrix expressions are sufficient.

    I wonder whether all other arguments using Ellerman’s dual logic and others are similar bluff of merely showing author’s wide range of knowledge. I do not see in this paper any real arguments on complexity that a human agent faces. Construction of a complex formal language must not be a part of complexity economics.

    • Frederico Botafogo says:

      Thank you, Prof. Shiozawa, for your critical comments. You raise important questions that require being addressed. As I understand it, you are asking two basic – and interrelated – questions: (i) Where is the complexity? (ii) Why do we need the additional math (tensors)? In what way does this additional math contributes to clarify the matter of complexity in economics?

      The answer to (i) is this: there already exists a body of knowledge that deals with complexity in economics; it is called accounting! Think about it: both when doing managerial accounting and when reporting their activities, firms do not take decisions based on production functions; they do not see their client firms or their consumers by reference to utility functions; they know they live outside of equilibrium such that a major issue is setting, and continuously adjusting costs and prices; past history provides valuable information about the immediate future, however it is accepted that the long term is simply unknown; etc. Accounting, or more precisely the double-entry bookkeeping procedure, is the sole underlying rule that frames all the above.

      This helps with answering the second question: if our goal is to mathematically frame the language of accounting, then we need to formally express the accounting equation in such ways that it allows addressing the issue of price and cost assignments. The accounting equation yields the balance sheet identity wherein numbers express ‘values’. Mathematically, value is very clearly defined as quantity times a number, I call it the p-number, which is interpreted as either some sort of price or some sort of cost. The tensor I introduce is required to convert the additive accounting equation in terms of values – assets equal liabilities plus equities – to a multiplicative accounting equation where q-numbers times p-numbers yield the neutral element 1, or 100%.

      I have read another comment of yours, which seems to indicate your appreciation for the work of Prof. Ellerman. He has a paper where he formalises his property theory using Graph Theory. Graph Theory is not a branch of mathematics most economists would be familiar with. He found the need for introducing new maths because he needs to account for cycles. My contention is that if you are out of the neoclassical paradigm, with its imposed equilibrium requirement, then you need cycles to account for value, prices, and money. Ellerman’s graph representation allows one to discuss the state of the whole economic system. However, I found the formalism in my paper, which relies on linear algebra, do be better suited for that same purpose.

      Why? Because I am framing teleological processes. Simmel, in his Philosophy of Money, seems to be a forerunner of this idea: in chapter 3, Money in the Sequence of Purposes, he writes

      ‘The great antinomy in the history of thought—whether the contents of reality are to be conceived and interpreted in terms of their causes or their consequences (i.e. the opposition between a causal and a teleological approach)—finds its original expression in a distinction within our practical motivations.’

      The consequence of the teleological approach is that we need to account for physical transformation (something you are right to say only requires vector analysis) alongside valuation assessments that express physical objects and physical processes in terms of the economic perspective. In my framework, q-numbers are quantities that are known as endowments, while p-numbers are valuations (prices or costs) that are desired at some future time and in such way that a firm can earn of profit.

      I guess this is my initial response. Should you want to continue this conversation, I am all in: please comment back or let’s engage directly – my email is

      I have three additional minor comments: as mentioned in the introduction to my paper, I was (and still am) unsure of the readerships’ background. Is there a need for the appendix wherein I review basic linear algebra? I don’t know. I am glad to find out that Prof. Shiozawa is well versed in this particular math.

      The bras and kets I introduce are not exactly the same objects originally developed by Dirac. Although my bras are linear operators on the kets, they also satisfy the constraint that bra times ket equals one: this is where the tensor pops up. I far as I know, Dirac was not dealing with tensors.

      The issue of partition is key to accounting: whereas economists claim they know what a commodity is, accountants introduce names to identify resources that are process-based such as ‘non-current equipment’ or ‘in-process inventory’. Thus, a car for an economist is just that, a car; for an accountant, it can be current inventory being hold by a used-car-dealership or a non-current equipment that will be useful for productive purposes over several periods of time. I claim reference to Ellerman’s partition logic is the way forward to properly deal with aggregation in economics, particularly when outside equilibrium and thus addressing issues such as the Cambridge controversies.

      Prof. Shiozawa, let me tell you that I’ve read your paper. I am glad to have identified some points of agreement, as follows:
      1. The underlying, foundational concept we are both trying to frame is ‘value’;
      2. We both work outside the neoclassical formalism; we rely on the classical framework, although it is challenge to be clear what we retain from that framework, and what we want to discard;
      3. I align with the view that value is a subjective construct, although it can gain ‘objective’ features by means of inter-subjective agreement among agents in an economy – typically, by reference to the concept of coalition of agents.
      4. Given 3 above, I trust it is possible to develop a description about how prices are bidded and offered in financial markets. The key is understanding agents as intermediary, who engage in trading for speculative purposes.
      5. In short, we are looking for micro-foundations to economics.
      6. What seems to put us apart (but I am sure this is not an un-surmountable obstacle) is that I rely on the accounting equation as the basis of any mathematically viable approach to modelling economics.

  • Yoshinori Shiozawa says:

    I have to say thanks for reading my papers. Points 3 and 4 are far from my under standing. I wonder why this kind of things happens. Probably I am a very bad presenter of my ideas.

    Accounting is important and complexity is a crucial factor in understanding why accounting plays indispensable role for management. But I cannot see any relevance of your talk about accounting. It seems that you are satisfied that you have connected mathematics and accounting in a formalism. Have you ever reflected why the collection of past data can contribute to the management? How do you explain it is necessary to reduce information by taking sum of numbers of a certain category (classification) obtained on various occasions? You can speculate on formalism but it seems you are missing the real questions concerning our capacity of understanding and the complexity of management problems. There are many points that we should reflect on the theme of usefulness of accounting. A mere formalism does not permit us to understand what we are doing in the name of management. Please read my old paper below:
    Shiozawa, Y. 1999 Economics and accounting: A comparison between philosophical backgrounds of the two disciplines in view of complexity theory. Accounting Auditing & Accountability Journal 12(1):19-38.

    As for formalism that you are developing, my friend Hiroshi Deguchi has already developed the theory of exchange algebra. It is not a mere argument on abstract algebra but contributes in reconstruct the SNA (System of National Accounting) from bottom-up point of view. See
    Deguchi, H. 2013 Economics as an Agent-Based Complex System: Toward Agent-Based Social Systems Sciences, Springer.

    Chapter 5 treats exchange algebra and Chapter 6 gives how it can be used in the re-construction of the SNA.

    • Frederico Botafogo says:

      Prof. Shiozawa,
      You are not a bad presenter. I guess this is my fault as I was writing a reply post: point 2 refers to the classical paradigm and as you appropriately argue, this is not a well-defined paradigm; the possibility of value being a subjective construct is mentioned in your paper; indeed, you do not align with this view; I was just trying to provide you with a reference by informing that I do; thus, this is how I would eventually deal with a question you ask at the very end of your paper concerning bidding and offering in financial markets.

      Mathematics is just a language such as English or Japanese. However, just as natural language are embedded within a cultural context, so is the APPLICATION of mathematics to any real-world problem. My point is that the mathematics of neoclassical economics FRAMES how we present, understand, and address any economic problem. The formalism is not neutral. Thus, with this ‘tentative’ working paper I am trying to introduce the formal syntactic elements of a language that supports dealing with accounting, economics, and complexity. To address your concerns, let me first read your old paper and I will come back to you in due course.

      • Frederico Botafogo says:

        Prof. Shiozawa, apologies for the delay in answering your last comment. Down here in New Zealand students are now graduating and I got tied up with some service duties.

        I now have read your paper. I am in full agreement. I see myself as pursuing your concluding remark that ‘… there is a promising field for interdisciplinary research and collaboration [between economics and accounting].’

        A great part of the paper informs the accounting readership how neoclassical economics provides a poor scientific framework for economic analyses. Given the background of WEA’s associates, this is a given; I have nothing to add except the comment that when one rejects utility and production functions, one must reject as well, as a matter of consistency, the idea of a market for each commodity.

        You then propose the idea of economic behaviour being framed by habits. This is an idea that fits well with accounting. I find your various discussion around management accounting to be spot on. The key issue I relate with is your addressing the need to reconcile ex-ante with ex-post analyses such that this exercise is logically inconsistent with maximization – and in fact, as you well noticed, it is inconsistent with simply making comparisons among alternatives (since alternatives may not exist, being developed until they are discovered, or they are not known).

        Within the discussion, you mention Yoshida’s CD-transformations. This is the bit I would have developed further. I am under the impression this is where a connection can be established between the overall discussion and the formal framework I propose. Further, my suggestion out of the traditional view for rationality (actual or expected utility) is not bounded rationality but logical consistency. If a decision-maker, think of a firm, knows or expects an undesired outcome, then rationally it should not proceed with the current course of action but rather should change course to whatever is perceived as being better. And looking backwards, if a course of action did end with an undesired outcome, then the firm rationally should acknowledge this for the sake of ‘see, control, and adjust the course’. The accounting equation can be framed as a mathematical statement about such consistency requirement. In my paper I present it as connecting present resources to future outcomes such that the process linking these two states must be consistent with the decision-maker’s knowledge (actual or subjective).

        Allow me some additional comments about my paper: all the points I make, including Ellerman’s partition logic (a point you have previously raised), are necessary for the proper framing of the accounting equation outside the neoclassical, general equilibrium, paradigm. They are not sufficient, however. Money and the proper notation for multi-agents simulation still need to be introduced. It is clear to me, now, that I failed to properly motivate the need for this formalism. If you give me the benefit of the doubt, I would be glad to argue how these points fit together to solve the puzzle of complexity within accounting. However, I am under the impression you made your mind that there is too much formalism for little content. Should that be the case, please wish me luck.

  • Ping Chen says:

    Botafogo made an important proposal by introducing accounting approach into complexity economics. I do not know his background is economics or accounting. I will raise the starting assumption of constant total value is not possible in real economies. Why? Firm’s costs and profits are varying depending on changing sale volume and production scale, since increasing returns in production or sales will lower average costs. There is also a significant time-delay and uncertainty between production and consumption, which is the roots of so-called accounting error or time-massage in financial and tax report. Neoclassical economics ignores this fundamental issue in accounting by simply assert equilibrium rate of profits is zero. In empirical statistics, profits rate across industries and across business cycles are changing and never converge to any constant or equilibrium rate. That is why linear matrix cannot approximate linear systems of dynamical differential equations. Time-varying quantity and profits are major source of empirical COMPLEXITY in ACCOUNTING.
    It the author replaces the constant TOTAL VALUE by RESIDUAL VALUE, which could be a useful indicator of positive/negative BUSINESS CONDITION if all input data from production to sales are EMPIRICAL DATA, then both management team and government authority may have a useful tool in firm management and macro policy. If the author wish to project future trend of cash flow under changing business condition, CAS/BAS model could run computer simulation with different scenario under changing technology and international trade.
    I wish the author could make breakthrough in accounting approach of complexity economics.
    Ping Chen, Fudan University and University of Texas at Austin, at 21:27, Nov.30, 2017, US Central Time.

    • Frederico Botafogo says:

      My interpretation of Prof. Ping’s comment suggests he has truly engaged with the paper and having done so he raises the key basic issue, namely how can a conservative system be used to model real-life, empirical economies? This is the question already asked by Gallegati, Keen, Lux, & Ormerod, (2006) in their paper title ‘Worrying trends in econophysics.’ Prof. Ping’s answer is that this is not possible. Once this answer is accepted as the starting point for analysis, I cannot but agree with his comments. However, I claim there is another possible answer.

      All economic data are framed by accounting. Within (double-entry) accounting, all transactions are conservative. The cost of inputs is always equal to the cost of outputs. When a sale takes place, the difference between cost and price is seen as a measure of profit. For the sake of argument, let us assume a deterministic setting wherein one knows how much profit is embedded within each sale. Then, the double-entry accounting will account for the conservation of costs plus profits. What are we to call this quantity? I suggest ‘value’ which as such would comprise ‘income’. Please see Fukui (2011) and Saito & Fukui (2015) for the resulting conceptual nature of ‘income’.

      The next point is moving from the single firm to the whole economic system. Accounting deals with aggregation; depending on the context, it may be also referred to as ‘consolidation’ or ‘business combination’. The formalism is such that one can combine all agents in an economy, both firms and consumers, to yield the balance sheet of the whole. There is no point doing this, except to perceive that within this formalism, increases/decreases in sales or costs will not change ‘value’, which remains constant.

      At the end of the day, my contribution is concerned with how best to conceptualise ‘value’ and ‘income’. As I said in the paper, I was and still am not sure how to introduce my ideas. Should Prof. Ping be available and allow me so, I would like to contact him directly to discuss how the linear framework presented here can replace the need for dynamic differential equations. The answer relates to Davis’ paper being presented just above mine.

      Gallegati, M., Keen, S., Lux, T., & Ormerod, P. (2006). Worrying trends in econophysics. Physica A: Statistical Mechanics and its Applications, 370(1), 1-6.
      Fukui, Y. (2011). The Imagined Dichotomy of Accounting versus Economic Income Concepts, Accounting, Economics, and Law: Vol. 1: Iss. 2, Article 6.
      Saito, S., & Fukui, Y. (2015). Whither the Concept of Income? Accounting, Economics and Law.

      • Ping Chen says:

        This is the MOST INTERESTING discussion in this forum. My answer is based on lessons in physics. For example, classical mechanics and quantum mechanics are all LINEAR in basic framework, but may introduce NONLINEAR INTERACTIONS in the description of FORCES or Hamiltonian function. In non-equilibrium statistical mechanics and non-stationary time series analysis. We do not have GENERAL EQUILIBRIUM,
        but we can introduce LOCAL EQUILIBRIUM as a MOVING approximation within a TIME WINDOW.
        It is known as SMOOTHING technique in accounting in deal with business cycle. For example, the manager of firm may save partial profits or inventory from THIS year for NEXT year in case of cash flow crisis. This practice may generate accounting error and even macro regulation. From my observation, China’s local governments also have incentive to SMOOTH its GDP growth rate and budget surplus/deficit in managing Central-Local or regional issues in economic development. In this sense, China’s economic system is more flexible than Western system in tax and accounting practice in dealing with business cycles and environmental crisis. Your accounting-based economic complexity has more fundamental impact to policy makers than technical-based complexity economics in REAL WORLD. Congratulations!