Money Theory

Shortest summary of the money theory from MoMaT-A, the axiomatics of economics.

Money theory cannot be built on double-entry bookkeeping (DEB). The central invariance of economics only appears in quadruple-entry bookkeeping (QEB), between two double-entry books. The sum of debtors equals the sum of creditors. From this follows the geometry of economics. It forms a coordinate system whose unit is the unit of bookkeeping (as the unit of an adjunction); with it the economy can be coordinated. The symmetry is this: a T-account balance in one DEB must correspond to another T-account balance, namely the one in the DEB of the contracting partner, on the other side of the balance-sheet identity (asset and liability accounts). From this symmetry follows an invariance — a Noether theorem of economics. This axiomatics is Riemann-equivalent: it is the smallest, universal system through which all behavioural theories of economics must factor. It thus spans the space of all bookkeeping-consistent behavioural theories — like the Riemannian manifold on which every physical theory must operate.

The unit of the coordination system is cash, an endogenous unit — unlike in physics, where the units of nature are exogenous. In the modern financial system (since the bill of exchange), demand deposits and reserve deposits are defined as units on this basis. This can be formalised through type theory, the mathematical theory of units. A debt contract opens a debt; performance contracts close it again — following the principle of abstraction and separation of the German Civil Code (BGB), through the terminal object cash. Debt contracts are, for example, a purchase contract, a credit contract, a deposit contract for demand deposits at a bank, or a reserve contract of a bank at a central bank. Opening the debt relation corresponds to differentiation — in Connor McBride's sense, the derivative of a functor, hence of a data type or a unit. It appears as a funding gap or a consumption of resources, initially without any cash flowing. Closing corresponds to the integral, hence to a categorical coend. It integrates over all agents of the economy — the credit and indebtedness layers of a labour- and risk-divided economy: households, firms, the government sector, banks, the central bank. This integrates out all opposing debt relations: clearing before settlement.

This identifies the Fundamental Theorem of Economics: differentiation is the inverse operation of integration. Differentiation opens a funding gap, integration closes it finally — for only cash, or central-bank money, can extinguish a monetary debt. This corresponds to the institution of money. It was fixed, together with the unit of the adjunction, for coordination, and it is a fixed point of risk prediction. An institution is a rule of the game being played, and with it the uncertainty about these rules is reduced. Within the institution, theory — the belief of an individual — becomes model, hence reality. General equilibrium theory thus confuses two uses of the fixed-point theorems of set theory: the one meant to explain dynamics, and the adequate one as fixed points of institution formation, where theory and model then coincide. As an explanation of dynamics this tool fails, for the difference between theory and model is exactly what constitutes the learning of agents and of the system. In institution formation, by contrast, it is adequate: there theory and model are meant to coincide — that is the very point of institution formation, the meaning, the model.

A current account at a bank is a right of instruction — to withdraw cash or to transfer the right of instruction (a transfer). Banks' reserves at the central bank are structured the same way. They serve above all clearing and settlement in the banks' liquidity pool, hence interbank trading via the central bank, which is a tool of the banks. A central bank is a liquidity pool of the banks and, as the apex of risk absorption, the institution that insures distressed banks. A consumption of resources without yield it books as consumption of society and distributes it over time into the future.

Cash is accordingly the terminal object of the money types: from it no further reference morphism leads to another money type. Its role is that of a balance-settlement standard or, in legal terms, that of the means for performance contracts. This structure also shows: whoever controls the issuance of cash also controls how what is produced together is distributed. Double-entry and quadruple-entry bookkeeping are, algebraically, universal algebras equipped with a theory and a model. In economics they additionally require a valuation algebra. In it, financial and real equity can be distinguished. This allows a gauge theory of economics to be formulated: the coend integrates out the financial equities as interaction balances in the QEB. What remains are the real assets, which express prices in monetary units. Inflation is thus explained not, as with Friedman, as a purely monetary phenomenon subject to central-bank policy. It is rather a phenomenon of the prices of input factors, hence of resources and labour.

The valuation algebra finally also ends the debates that followed Marx — the debates over whether the capitalist (the entrepreneur) exploits the workers. Marx's question of where profits come from is very important, but his answer was wrong. Profits come from the parallelism and the temporal divergence of the investment credits, the financial endowment. This financial endowment complements the real endowment of Walrasian and general equilibrium theory. They are what makes a division-of-labour economy possible in the first place. Without this initial financial endowment, a labour- and risk-divided economy cannot produce goods from its real endowment of resources and labour. Risk-sharing is necessary because the input suppliers — labour, resources and land — cannot or will not bear the risk of producing and selling the goods. This risk is borne by the entrepreneur and the banks. Dually, employees, landowners and resource owners receive a salary or rent every month — regardless of whether the firm succeeds, with the secondary risk that their source of income is rationalised away. In the international process of sharing labour, risk and profit, valuation algebras make it possible to explain and secure the sovereignty options of the national central bank transparently from the bookkeeping.

Marx's diagnosis should not have been exploitation, but the engineering of a societal innovation, that of the bill of exchange. In the third volume of Capital there are indeed ideas on the polymorphism of the clearing and settlement process by means of the bill of exchange. Marx and Engels could, in principle, have identified the bill of exchange and the clearing and settlement processes accompanying it as the fundamental structures of capitalism. This is what we have carried out here — with the axiomatisation of bookkeeping and the modern tools of the semantics of computability theory, the hylomorphisms.

Engineering arises when a science transitions into a large-scale construction technology — when the Fundamental Theorems become compositional. In physics, Lagrange achieved this with the Lagrange equation and the Lagrange multipliers. In economics these are used, but they have so far not been built up into the compositionality of economic theories. We have carried this out with Open Games, a theory of compositional agents committed to logical AI. In this axiomatics we underpin that theory with the data structures of the economy, hence with bookkeeping. In the following step we develop it into a hierarchical decision theory — economic policy.

A science arises when an applied mathematics moves from a phenomenological to a semantic tool — when one defines the units and formalises their emergence. In economics both are still outstanding, and we deliver both: a theory of units and a theory of composition. This makes economics applicable — and, in our case, programmable. The categorical formulation is a language with which economic policy can be specified. The compiler technology of category theory makes it executable, as software for the information processing of the economy.

Differentiation (the funding gap after debt contracts) and integration (closing through performance contracts) are mutually inverse. They reveal the polymorphism of clearing and settlement operations at all levels of the economy: within a firm, differentiation is a cost centre; in the supply chain, the funding gap; in the currency area, the international current and capital accounts. Our money theory is thereby a cost-and-revenue accounting — the profit identification that Marx did not see — and a profit-distribution accounting. Our compositional money theory also makes operational cost-and-revenue accounting compositional: the accounts of the subsidiaries compose into the account of the holding. Restructurings, too, can be simulated this way — only through the compositionality of the newly structured divisions.

The engineering of economics, which we formalise in our axiomatics, establishes: the dynamics of a labour- and risk-divided economy follow a hylomorphism, hence a composition of an algebra after a coalgebra. The coalgebra unfolds production out of the investment, hence out of the initial financial endowment. This appears as the firms' supply of goods and as their demand for money to repay the credits in order to remunerate the input factors. The algebra folds the supply of goods into consumption. Aristotle's teleology, the purpose of the economy, thus emerges: the entrepreneur anticipates his turnover. If the turnover then realises, the fundamental learning unit of capitalism arises. Demand validates the risky investment. Keynes exposed saving for what it is: a refusal of validation, an interruption of the debt-extinguishing cycle. When the state steps in, things usually do not go well. For it does not validate with its own money, a principal–agent problem arises, and this money is often debt-financed and not repaid. Ultimately this is a diversion from private to state use of resources.

Of course, the financing and production of public goods follows a different economic rationality than closing private demand gaps for private goods. The management of public goods within Elinor Ostrom's polycentric management structures is structurally possible with our compositional economics, and the threefold nature of her analytical tools corresponds to our threefold topos of governance, decision and accounting. Compositionality then yields polycentricity. Capitalism has so far been a natural system of labour- and risk-sharing. With the valuation algebra we now also have the tool to compute the profit-distribution function. This allows answers to the questions of cost-and-revenue accounting: who gets what when work is done together but risk is shared?

The initial algebras are the data structures, the final coalgebras the dynamics of the economy. Both are functorial fixed points and answer the central economic-policy requirement on monetary systems: that of their stability. The initial algebras of the gold standard are not stable and are inherently exposed to speculative attacks. Paper money is inherently stable, hence resistant to speculation, because the central bank can issue it without limit. It thus provides, for any quantity of debt contracts, the means of de-indebtedness — that is, the settlement means for the performance contracts: cash.

This is the shortest summary of our money theory for the axiomatics of economics. To formulate it, we use the most abstract mathematics that currently exists: category theory, used for the semantics of programming languages. The requirement was to deliver not only a money theory but also the specification for implementing its software systems. Category theory is thus the normal and only viable path — because it is the path of computer science developed for exactly this — to meet that requirement. It currently also serves as the semantics of mathematics itself, for in topos theory it allows the identification of structures across different mathematics — hence the preservation of structure under the transformations between them. These internal mathematics are therefore the proper place — the topos, where observer and observed can be situated simultaneously; here the Lucas critique can be dissolved, which is exactly what characterises social theories. Put differently: only here do we find Aristotle's telos and the space between Plato's cave and nature — the space in which human-made structures live: money and the institutions of our cultures, made by and for humans.