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In order to effect control of any system we must first have data on that system. A planned economy, such as I've been writing about for the last few years, is synonymous with economic coordination in kind. For such coordination to not be completely arbitrary, to devolve into subjectivist nonsense, we must have accurate in-kind data. Accounting in kind, accounting in physical terms, can be used to gather such data. What we do with the data once gathered, and what data is sensible to gather in the first place, can be debated. But what is certain is that we cannot provide guarantees that the human economy is kept within environmental bounds, while fulfilling human needs, without having some idea of what each part of the economy is doing. In-kind accounting can be used not only to track production, but also for inventory, transportation and for detecting theft. The unloading of a container from a ship is an accounting operation, as is the further transport of that container. The container is "created" when it is packed at the source and "destroyed" when it is unpacked at the destination. As accounting maneuvers the creation step can be booked as crediting a "containers in flight" account and debiting the container to wherever it will picked up from. Once the container is unpacked at its destination it is credited from the destination and debited to the in-flight account, making all accounts associated with the container sum to zero. As long as the container is in flight it appears on the books as a liability. If the container is lost, whether by accident or by theft, that too can be booked as a transaction to some appropriate account. When workers at a warehouse perform inventory, any discrepancies between expected and actual inventory can be accounted for by debiting and crediting relevant accounts. For example, if the books show that there should be ten tons of gold in a vault but there are only nine tons, then this can be booked by crediting the warehouse one ton of gold and debiting a discrepancy account one ton of gold. Obviously such a discrepancy would be very serious and grounds for investigation, but it should first of all be accounted for. Another example could be a tank car of methylamine showing a mass discrepancy between source and destination. Unusually high or low yields in certain manufacturing processes could also be detected, by performing statistical analysis on the in-kind accounting data. Some time after I started writing this post I was sent a 2007 paper by David Ellerman [3] that talks about vector-valued in-kind accounting. I will use Ellerman's terminology where appropriate. Ellerman's analysis To get us started I will repeat what Ellerman says. T-accounts and the Pacioli group Accounts in traditional accounting are called T-accounts , due to their table borders often being drawn in a way that resembles the letter T: Debit Credit d c Ellerman points out that traditional double-entry accounting forms what in mathematics is called a group [3] , specifically a so-called group of differences. He names this group the Pacioli group after the Franciscan friar Luca Bartolomeo de Pacioli, who first described double-entry accounting in his 1494 text Summa de Arithmetica [6] . Ellerman chooses the symbol P for elements of this group, which is equivalent to N 0 2 , the set of pairs of non-negative integers. Following Pacioli, he denotes elements of P like this: [ d / / c ] where d stands for debit and c stands for credit. Ellerman names these elements T-terms . Ellerman defines the identity element of P as [ 0 / / 0 ] . He defines addition pairwise like so: [ w / / x ] + [ y / / z ] = [ w + y / / x + z ] Equality is defined by what Ellerman calls the cross-sum: [ w / / x ] = [ y / / z ] ⇔ x + y = w + z Finally, Ellerman defines the inverse of an element of P as simply swapping c and d , and shows that adding the inverse of an element to itself results in the identity element, which is a must for an abelian group: [ y / / z ] + [ z / / y ] = [ y + z / / y + z ] = [ 0 / / 0 ] Ellerman calls any T-term that is equal to [ 0 / / 0 ] a zero-term , and points out that any balanced transaction is a zero-term. A journal is a list of such balanced transactions, the sum of which is also a zero-term. Ellerman points out that the "double entry" in double-entry accounting doesn't refer to the two columns debit and credit, but to the fact that any modification to an equation must affect at least two terms of that equation if the equation is to hold. This observation goes back to at least Muhammad ibn Musa al-Khwarizmi, from whose work al-Jabr (completion/rejoining/balancing) we derive the word algebra . The concept is likely as old as mathematics itself. The double entry principle applies also when using a single column of signed values to record things. Compare the following two methods of recording a transaction of 1000 units from account Bar, and 250 units from account Floop, to account Foo: Account Debit Credit Bar 0 1000 Floop 0 250 Foo 1250 0 Account Amount Bar -1000 Floop -250 Foo 1250 In the first table we need the sum of all entries in the debit column to equal the sum of all entries in the credit column. In the second table we need the sum of all amounts to be zero. Ellerman points out that either approach is valid, and that they are in fact equivalent [3] : The real choice between the double entry method and the complete single entry method of recording a transaction is the choice between using unsigned ("single-sided") numbers in two-sided accounts or signed ("two-sided") numbers in "single-sided" accounts. Multi-dimensional accounting In a 1986 paper Ellerman extends the above concept to vectors of non-negative integers ( N 0 n ), and more generally to vectors of non-negative reals ( R ≥ 0 n ) [2] . The formulation is entirely the same as the scalar case, only with vectors instead of scalars. For example, the identity element is defined as [ ( 0 , 0 , … 0 ) / / ( 0 , 0 , … 0 ) ] . Taking the previous example, let us pretend it concerns the sale of some gizmo at a price of 1000 + 25% VAT. Ellerman calls the money and gizmos two properties that are to be accounted. Let us extend the example to two-dimensional vectors with properties ( Money , Gizmos ) , using Ellerman's notation: Account [Debit // Credit] 3001 Sales [(0, 1) // (1000, 0)] 2610 Outgoing VAT, 25% [(0, 0) // (250, 0)] 1930 Company account [(1250, 0) // (0, 0)] 9001 Inventory [(0, 0) // (0, 1)] The four-digit numbers are taken from the standard Swedish account plan (kontoplan). The entire 9XXX range is reserved for internal accounting, so I use that for inventory. I have chosen to account the debit of the gizmo to the customer via the sales account. We can see that the sales term represents C-M exchange (commodity-money). The entire transaction sums to [ ( 1250 , 1 ) / / ( 1250 , 1 ) ] , which is a zero-term, as required for a transaction. Single-sided multi-dimensional accounting Ellerman doesn't go into single-sided (signed) accounts in his 1986 paper, so I will demonstrate the concept here. By subtracting credits from debits we arrive at the following: Account Amount 3001 Sales (-1000, 1) 2610 Outgoing VAT, 25% (-250, 0) 1930 Company account (1250, 0) 9001 Inventory (0, -1) In the above it isn't immediately obvious what is being accounted for. By moving the properties into the table header and splitting them up into two separate columns we get the following: Account Money Gizmos 3001 Sales -1000 1 2610 Outgoing VAT, 25% -250 0 1930 Company account 1250 0 9001 Inventory 0 -1 I find this far easier to read compared to Ellerman's notation. With double-sided accounts it would look something like the following instead: Account Debit Credit Money Gizmos 3001 Sales 0 1 1000 0 2610 Outgoing VAT, 25% 0 250 0 1930 Company account 1250 0 9001 Inventory 0 1 The benefit of the above is that it is closer to what traditional accounting looks like. But it is also more verbose. Personally I prefer the single-sided (signed) version. Non-exchange The exchange of gizmos for money is value-preserving, assuming general equilibrium. In an economy without exchange, for example a gift economy, value is not preserved. If for example, instead of selling the gizmo, it is simply given to other entity, the transaction simplifies to: Account Gizmos 9001 Inventory -1 6993 Gifts 1 If both entities do their accounting in a common system then the transaction may look like this: Account Gizmos 9001 Inventory A -1 9002 Inventory B 1 There is no counter-flow of value here. The transaction is akin to a requisition. This is the case with the free gifts of Nature, in all modes of production. Nature should therefore have an account in an in-natura accounting system. In contrast, Ellerman says the opposite [3] : For example, production is often viewed as an exchange with Nature where the inputs are the property given up to Nature and the outputs are the property acquired back from Nature. This metaphor may be helpful for illustrative purposes, but 'Nature' will not be awarded an account in property accounting. But if for example we dig up coal, that coal has to be credited from somewhere in order to enter into the in-kind accounting system. It therefore makes perfect sense to credit Nature for it. An account for socialist primitive accumulation would likely also be necessary for similar reasons. The historical reason for unsigned accounts Traditionally accounting is done using double-sided unsigned accounts, as explained above. I suspect that this method comes about for historical materialist, technical reasons. It is much easier to sum two columns of unsigned numbers separately and then computing balances, than it is to sum single columns of alternating positive and negative numbers. This is the case when adding numbers by hand, but also when using certain adding machines. Only with the advent of full computerization has the mixed adding of positive and negative values stopped being much of an issue. The material need for unsigned accounting has therefore disappeared. Rational-valued accounts In Ellerman's analysis accounts are either integer-valued or real-valued. While we in theory could transfer π units of something between accounts, such accounting requires symbolic manipulation. This is impractical, so I will not consider it further. There is however a practical need for non-integer accounting, and I will make the case here that arbitrary precision rational numbers can meet this need. Consider accounting grain shipments. While in the best of worlds everyone would use the same system of measurement, be it metric or imperial, in practice there is a mix of measurement systems and devices in use. If we used integer values, agreeing on some common denominator in the system, then as soon as an amount of grain measured using a system which doesn't neatly fit into the agreed-upon denominator, that entry would either have to be quantized int