An Inconsistency in using Stock Flow Consistency in Modelling the Monetary Profit Paradox

In order to understand from where the profits or monetary profits of capitalists and firms emerge I examined the phrase of Marx, ‘Die Gesamtklasse der Kapitalisten kann nichts aus der Zirkulation herausziehen, was nicht vorher hineingeworfen war.’ (The class of capitalists cannot extract from the circulation, what has not previously been thrown in.) Also Keen studied the monetary paradox and contrary to circuitists he came to the conclusion that capitalists can make monetary profit with a possibility to earn enough to repay their debt and with positive balances for all actors. I will prove that Keen made a fundamental mistake and is using the Stock Flow Consistency Principle in an inconsistent way by combining it with behavior equations in a dynamic model. So the solution presented here is not only showing that the numbers are incorrect but the method itself. This resolves a contraction between Keen and circuitists and implies that, in a Wicksellian pure credit economy, it remains impossible to gain a monetary profit for all actors. More precisely, that the total sum of monetary profit over all actors is zero.


Introduction
In its most simple form the paradox is described as: A capitalist can put workers to work against wages W. The workers buy the consumer good from the capitalist for C=W. And then the question arise how can the capitalist make a profit from it. For an extensive historical overview on this topic see Tomasson & Bezemer (2010).
In a discussion paper Keen (2010-1) tried to solve the monetary paradox too and came, in contrary to circuitists, to the conclusion that capitalists can make monetary profit with a possibility to earn enough to repay the debt and with positive balances for all actors. I will prove that Keen made a fundamental mistake and is using the Stock Flow Consistency Principle in an inconsistent way by combining it with behavior equations in a dynamic model. The solution presented here is not only showing that the numbers are incorrect but the method itself. This resolves a contraction between Keen and circuitists and implies that, in a Wicksellian pure credit economy, it remains impossible to gain a monetary profit for all actors.
In the published paper (Keen,2010-2) the numbers are different but the fundamental mistake is still there.

Analysis
We will follow Keen's reasoning step by step and show where his mistake took place. First Keen is building a small closed economy with workers, a firm and a bank. For the definition of the terms used I like to refer to De la Fonteijne (2013). Then he is adding a bank because the argument is that with fiat money each transaction is in principle a tripartite action between buyer, bank and seller. And another argument is that with a bank there is interest involved.
This process results in a stock flow table as Godley (2007) is producing in his Stock Flow Consistency (SFC) approach with corresponding differential equations as behavior functions .
The model begins with the banking sector extending a loan Λ to the firm sector; this initializes the system by creating Λ of credit money stored in the F D account, for which there is a matching record of debt in F L . The minimum set of flows that this creation of credit money sets in train is: 1. Accrual of interest (A) compounds the outstanding debt in F L at the rate (r L ) specified in the loan contract; 2. Assuming that the firm sector meets its debt-servicing obligations in full, a flow of money (also A) from F D to B I offsets the compounding of debt in the first operation; 3. A flow of money (B) from B I to F D pays the firm sector interest on its deposits at the rate r D a lower rate than that charged on debt; 4. A flow of money (C) from F D to W D pays wages to workers (who are then employed in factories to produce output for sale); 5. A flow of money (D) from B I to W D pays workers interest on their bank balances; and 6. A flow of money (E and F) from both B I and W D goes to F D to pay for the output from the factories owned by the firm sector.
Because F L is not changing I left it out of the table, so in fact it is a revolving loan.
These conditions can be specified more precisely by making simple constant parameter substitutions and introducing the following behavior: 1. A = r L Λ is the loan rate of interest r L times the amount outstanding in the loan account F L .
2. B = r D F D is the deposit rate of interest r D times the balance in F D ; 3. C = w F D is a factor w of the current balance in F D 4. D = r D W D is the deposit rate of interest times the balance in W D and 5. E = ω W D and F = β B I will be some factor (say ω and β respectively) of the balances in the accounts W D and B I Now we can solve for the equilibrium levels of these accounts, which will give us the conditions for F D , W D en B I .
Note that one of these three equations is redundant due to Walras law (Godley, 2007).
So solving these equations needs another not mentioned equation. Keen is using Because of his choice in using a part of the Bank balance this is not a very beautiful solution but not wrong either.
In equilibrium the solution is He is therefore introducing the following equation: Aggregate wages are therefore W = F D Since national income resolves itself into wages and profits Keen argues (interest income is a deduction from other income sources), we have also identified that gross profit П equals

and GDP Y equals
The problem is that this additional equation is resulting in a strong interdependence of the equations and parameters used. So once you choose τ s , with given w, than s can be calculated (iteratively).
As an example Keen let s = 0.3 and τ s = 0,25. E and F are the value of the sold product of the firm and this has to add up to GDP Y. The other approach is production costs plus mark-up which is of course also equal to GDP Y.
That leaves us with the question why E + F is not equal to Y if we are talking about the same system and again this is because the system has changed and is over-determined which leads to a contradiction. The system is no longer SFC. So it is not possible to repay in this case the initial loan in no matter how many years.

Conclusion: the proposed solution and conclusion are wrong
 From a mathematical point of view, because the introduction of the new formulas makes the system inconsistent except for s=.012506  From a fundamental point of view, because in a closed system the total of the financial assets cannot change, so ΔF D +ΔW D +ΔB I =0 (de la Fonteijne, 2013), which also applies to a Wicksellian pure credit economy.
 From a sustainable and philosophical point of view, because otherwise Keen would have created a monetary money making machine with unlimited capacity, which can be considered as a perpetuum mobile.  From an economical point of view, because you cannot increase Y as you please and because of the strange choice of modelling behavior on the opposite extreme of what Godley [2007] is using throughout his book.
For the reader who is interested in the unravelling of the monetary profit paradox I like to refer to my paper 'The Monetary Profit Paradox and a Sustainable Economy: A Fundamental Approach' (de la Fonteijne, 2013).