### CH Douglas A+B Theorem

JC 04 Jan 2018
CH Douglas' A+B Theorem (Social Credit) has always been opaque (to me), and discussions / proofs have always seem to rely on proof by rant. This may be due to the lack of a clear time-dependent model.

The import of the A+B Theorem is that an economy without a supply of new credit is unable to distribute sufficient payments for the output of the economy to be purchased. Most commentators have found this theorem to be wrong.

Here, I use a simple time-dependent model based on Steve Keen's paper: "Solving the Paradox of Monetary Profits" (2010) of a very basic economy that uses a fixed quantity of notes as a lubricator, but without any creation of credit.

A simple model of a basic economy

Consider a household which owns an apple orchard and that decides to go into business. They hire workers to expand the orchard, look after the trees, pick and sell the crop.

The household needs to pay the workers, and to do so they engage the services of a friends who have a Monopoly set.

The friends set aside M 100 of Monopoly notes and put them into a piggy bank: this will be the source of all funds for the basic economy - no more notes will be added nor subtracted. These friends call themselves "The Bank".

The Bank agrees to supply notes to the Orchard Firm at an interest rate of 5% per annum (5% pa).

The Bank in fact sets up two piggy banks - the one that holds the original bank notes (The Vault), and another to manage moneys coming from interest payments (Bank Assets).

The Firm sets up two piggy banks - one to hold the loan from The Bank, the other to manage moneys coming from the sales of apples (Firm Deposit Account).

The Workers set up one piggy bank (Workers Deposit Account) to receive payment for their work.

The parameters of the model are shown here:
 Bank Vault starting value M 100 Bank Vault lend rate 75% of Vault balance pa Loan repayment rate 2% of Firm's assets pa Workers payment rate 200% of Firm's assets pa Workers spend rate 26 x 100% of Workers savings pa Bank spend rate 100% of Bank Assets pa
(M is the (Monopoly) monetary unit of account; pa = per annum)

To explain these parameters:
• The Bank Vault holds only M 100 of notes and no more. To formulate a dynamic model, we need to choose a rate at which The Bank lends out notes in such a way that the Vault is never overdrawn. This model takes the rate of lending to be 75% of the current Vault balance per annum. Over time, other things being equal, this would give an ever decaying balance tending to zero.
• The Firm repays the Bank loan at a rate of 2% pa of its total assets (loan plus income)
• The Firm pays wages at a rate of 200% pa of its total assets (loan plus income). At first sight, this may seem impossible, but remember that this is a flow (amount per unit time) rather than a stock.
• Workers spend their entire paycheck every 2 weeks.
• The Banks spends its entire income each year.

Using Steve Keen's program Minsky, the above model can be simulated over time (Click on image on the right).

A more full explanation together with the equations of this model is here: Model equations

Initial values grow or decay until a steady state is reached after approximately 6 years.

The final (steady state) values are shown below (values in M)

Stocks
Bank Vault 16
Loan to Firm 84
Bank Assets 4.2
Worker Deposit Account 5.7
Firm Deposit Account -9.9

Flows (M pa)
Interest on Bank loan 4.2
Wages paid 148
Worker consumption 148
Bank consumption 4.2
Vault Lend per unit time 12.0
Loan repayment per unit time 12.0
The M 100 of Bank notes have lubricated an economy to produce 148 + 4.2 of consumption. 16 of the 100 remain in the Vault; 84 are on loan to the Firm.

It is interesting to note that the amount paid in Wages (148) per year is greater than the original number of Bank notes (100). This is due to the recycling of notes: the notes in circulation is 84 while the GDP is 152.2, giving a "velocity of money" of 152.2 / 84 = 1.8

The Firm's negative Deposit Account (-9.9) seems a bit strange. The Firm's Deposit Account balance equals the sum of the Bank Assets and Worker Deposit Accounts (4.2 + 5.7 = 9.9), meaning that the Bank and Workers are creditors of the Firm: they have been over-paid by one time unit. (In the Workers' case, this means 148 / 26 = 5.7).

Profit & Loss
Sales:
Worker consumption 148
Bank consumption 4.2
152.2
Expenses:
Wages 148
Interest on Bank loan 4.2
152.2
Profit 0

Balance Sheet
Assets:
Bank loan 84
Firm Deposit Account -9.9
Creditor: Workers 5.7
Creditor: Bank (interest) 4.2
84
Liabilities:
Bank loan 84
84

As it stands, this model does not appear to confirm the A+B Theorem that such an economy cannot function without an injection of credit from outside, in accord with the simple stock accounting model.