Saturday, 26 July 2014 02:54

A+B: A Mathematical Reply to an Objection

Written by Jim Schroeder
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In his book Credit-Power and Democracy, C.H. Douglas introduced the A+B theorem as follows:

 

"A factory or other productive organization has, besides its economic function as a producer of goods, a financial aspect – it may be regarded on the one hand as a device for the distribution of purchasing-power to individuals through the media of wages, salaries, and dividends; and on the other hand as a manufactory of prices – financial values. From this standpoint, its payments may be divided into two groups:

Group A: All payments made to individuals (wages, salaries, and dividends).
Group B: All payments made to other organizations (raw materials, bank charges, and other external costs).

Now the rate of flow of purchasing-power to individuals is represented by A, but since all payments go into prices, the rate of flow of prices cannot be less than A+B. The product of any factory may be considered as something which the public ought to be able to buy, although in many cases it is an intermediate product of no use to individuals but only to a subsequent manufacture; but since A will not purchase A+B; a proportion of the product at least equivalent to B must be distributed by a form of purchasing-power which is not comprised in the description grouped under A. It will be necessary at a later stage to show that this additional purchasing power is provided by loan credit (bank overdrafts) or export credit." [1]

 

Now, it is often alleged by critics of the theorem that the theorem itself is absurd because there are times when aggregate income is greater than or equal to the price of consumer goods coming onto the market. However, the theorem does not state that total incomes are necessarily less than the price of consumer goods. What the theorem states is that total incomes are necessarily less than total prices generated in the same period of time in all industries.

In mathematical notation:

1. Sum (∑) of total incomes < sum of total prices (for all time t>0 – or at least since the industrial revolution)

2. There are two types of goods, capital goods and consumer goods,

Therefore two types of prices: capital goods prices and consumer goods prices, and two types of incomes derived from production: incomes derived from the production of capital goods and incomes derived from the production of consumer goods.


3. Therefore, A+B theorem can be restated:

Sum (capital incomes + consumer incomes) < sum (capital prices + consumer prices) (for all t>0).


4. Consumer's aggregate income = Sum (capital incomes + consumer incomes).


5. Therefore, Consumer income can only be ≥ consumer prices

if Sum (capital incomes +consumer incomes) ≥ consumer prices.


6. Since capital prices in time t, always show up in consumer prices in t+x:

If Consumer income ≥ consumer prices in time t,

then

Sum (capital incomes+consumer incomes) time t+x < sum (consumer prices), unless capital incomes are increased
by an amount greater than sum (capital prices in time t).

 

What the theorem demonstrates is that if attempts are made to equate aggregate income with the prices of consumer goods through the production of additional capital goods or services, such attempts will necessarily lead to a point where aggregate income is less than the price of consumer goods at a future point in time. In other words, the gap between income and prices never disappears, but is only masked by the production of additional capital goods and will show up later in an aggrevated form.

 

Keynes recognized the problem with regard to this method of bridging the gap when he wrote:


"Thus the problem of providing that new capital-investment shall always outrun capital-disinvestment sufficiently to fill the gap between net income and consumption, presents a problem which is increasingly difficult as capital increases. New capital-investment can only take place in excess of current capital-disinvestment if future expenditure on consumption is expected to increase. Each time we secure to-day’s equilibrium by increased investment we are aggravating the difficulty of securing equilibrium to-morrow." [2]

 

1. C.H. Douglas, Credit-Power and Democracy (Melbourne: The Social Credit Press, 1933), 21-23.

2. J.M. Keynes, The General Theory of Employment, Interest, and Money (New York: Harcourt, Brace & World, Inc., 1964), 105.

Last modified on Saturday, 10 February 2018 17:36

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3 comments

  • Comment Link Jim Monday, 29 January 2018 02:18 posted by Jim

    Payments to other organizations do not necessarily represent income to individuals. In fact, except for the small amount of revenues distributed as dividends, payments to other organizations are used to pay debt or replace working capital. It is not distributed as income to anyone. Companies do not wait until the good or service that their workers create is sold before they pay those workers' wages or salaries. Most companies operate on a revolving line of credit and pay their workers far in advance of the good or service they produce being sold. When that good or service is sold, they then use that revenue to repay the debt. This is especially noticeable in capital production where the incomes distributed in the production of said capital is distributed years before that capital makes its way to the cost of some consumer good. The economist's theory is that these people will save this money in order to defray the cost of said capital, so that income = costs, but in reality, this money is used to purchase consumer goods on the market at, or near, the time of the income distributed. This increases the demand for consumer goods at, or near, the time of the capital's production which also tends to inflate the price of said consumer goods.

    In other words, consumers pay for capital twice. Once in the form of inflation of consumer goods, and again as that capital is depreciated over time. However, the income necessary to defray those costs is only distributed once. Capitalized costs merely represent past consumption.

  • Comment Link  Francis L. Goodwins Monday, 29 January 2018 02:17 posted by Francis L. Goodwins

    The issue here is really just the problem of how to account for period distribution of goods and incomes, and while correctly solved, doesn't throw much light on the real problem with CH Douglas' theory, namely that the division of the payments made in the production process into the two categories, A and B, is untenable. In the quotation above, Douglas attempts to distinguish between payments to individuals and payments to organizations. But organizations are made up of individuals, and since we live in a world where all real wealth (as distinct from money) is owned, the fact is that all payments ultimately reach individuals. Take raw materials. I can't conceive of how a payment for materials can be made that does not go to the individuals owning them. Bank interest becomes revenue of the banking organization, and ultimately arrives in individual pockets as salaries, dividends or profits (which I suppose have been concealed in "other external costs").

    But the fascinating thing is that Douglas identified a real problem with the economic system, its failure to generate sufficient consumer purchasing power to sustain equilibrium, so the question is, if the A+B Theorem can't explain the problem, what can?

  • Comment Link Steve Monday, 29 January 2018 02:16 posted by Steve

    Yes, and the only way to actually macro-economically solve this problem (as opposed to palliating it as Keynes suggested) is to implement a policy of giving individuals a supplementary income which virtually equates total individual incomes with total prices, and also implementing a voluntary policy of a mathematically derived general discount on prices by retailers after they have discovered/decided the prices of their various goods/services and which discounts are rebated back to those merchants. Thus an equilibrium is attained and maintained through time.

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