A+B: A Mathematical Reply to an Objection

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

     iff 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.